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Day-1
MOCK TEST-1
Conceptual Quiz on Divisibility and Prime numbers.
Number of Questions-> 100.
Sources-> The following questions are from various sources and quite a few are self made.
1)If z is divisible by all the integers from 1 to 5 inclusive,what is the smallest value of z?
2)Data Sufficiency-> Is the positive integer z divisible by 12?
A)z is divisible by 3.
B)z is not divisible by 2.
3)Data Sufficiency-> Is integer z divisible by 21?
A)z is divisible by 7.
B)z is divisible by 3.
4)Data Sufficiency-> Is integer z divisible by 48?
A)z is divisible by 12.
B)z is divisible by 4.
5)Data Sufficiency-> Is integer z divisible by 12?
A)z is divisible by 8.
B)z is divisible by 6.
6)Data Sufficiency-> Is integer z divisible by 24?
A)z is divisible by 6.
B)z is divisible by 4.
7)How many prime numbers between 1 and 100 are factors of 168?
8)How many Prime factors does 2180 have?
9)What is the greatest prime factor of \(49^{19}-7^{35}\)
A)2
B)3
C)7
D)13
E)19
10)If n is the product of integers from 1 to 12 exclusive,how many prime factors does n have?
11)If 2,3,5 are the prime factors of N,what is the value of N?
A)2
B)3
C)5
D)2*3*5
E)cannot be determined.
12)If 2,3,5 are the "ONLY" prime factors of K,what is the value of K?
A)2
B)3
C)5
D)2*3*5
E)cannot be determined.
13)Data Sufficiency -> What is the value of positive integer K?
A)2,5,7 are prime factors of K.
B)K<100
14)If p is a positive integer and 200 multiplies by p is square of an integer,what is the value of p?
A)2
B)3
C)5
D)10
E)Cannot be determined.
15)Data Sufficiency->What is the value of positive integer p?
A)200 multiplied by p is square of an integer.
B)p is an even number less than 15.
16)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is an even number less than 15.
17)10^25 – 560 is divisible by all of the following EXCEPT:-
A)11
B)8
C)5
D)4
E)3
18)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is a factor of 75
19)Data Sufficiency->What is the value of positive integer p?
A)200 multiplied by p is square of an integer.
B)p is a factor of 60.
20)Data sufficiency-> If a and b are prime numbers,what is the value of a*b?
A)a-b=1
B)a=3
21)If x and y are prime numbers, each greater than 2, which of the following must be true?
I. x+y is an even integer
II. xy is an odd integer
III. (x/y) is not an integer
A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
22)Which of the following statements must be true->
A)zero is neither prime nor composite.
B)one is neither prime nor composite.
C)-3 and 3 are both primes.
23)Which of the following statements must be true->
1)A prime number must be positive.
2)For any prime number p,there is no x such that 1<x<p and x is a divisor of p.
3)The product of first ten primes is even.
4)All prime numbers greater than 71 are odd.
5)2 and 3 are the only consecutive integers that are also prime numbers.
6)p is a prime number and x and y are positive integers.If p=x*y then one out of x or y must be one.
7)All the prime numbers greater than 3 can be written as either 4n+1 or 4n-1.
8)All the prime numbers greater than 3 can be written as either 6+1 or 6n-1.
24)Data Sufficiency->If p is a prime number,what is the value of p?
A)11<p<17
B)p is divisible by 13.
25)Data Sufficiency->If A and B are positive integers, is A/B an integer?
(1) Every factor of B is also a factor of A.
(2) Every prime factor of B is also a prime factor of A.
26)Data Sufficiency->Is \(\frac{n}{14}\) an integer?
(1) \(n\) is divisible by 28.
(2) \(n\) is divisible by 70.
27)If x is a prime number, what is the value of x?
(1) There are a total of 50 prime numbers between 71 and x, inclusive.
(2) There is no integer n such that x is divisible by n and 1 < n < x.
28)Data Sufficiency->If p = (n)(5^x)(3^k), is p divisible by 10?
(1) n, x, and k are even.
(2) x > n > k > 0
29)Data Sufficiency->If x is a prime number,what is the value of x?
1)There are 3 prime numbers between 11 and x
2 )x+1 is a prime number
30)If \(A\) is positive integer,then \(A\) , \(A^3\)and \(A^7\) will have the exact same prime factors.
A)True
B)False
31)Data Sufficiency->If y and x are positive integers, is y divisible by 3?
(1) y=x(x+1)
(2) y^2 is divisible by 9.
32)Data Sufficiency->If y and x are positive integers, is y divisible by x?
(1) y = x^2 + x
(2) x has the same prime factors as y.
33)Data Sufficiency->If z is a positive integer,is (z^31+7)^2 divisible by 4?
A)√z has five prime factors.
B)All prime factors of z^3 are greater than 7.
34)If x is a positive odd integer and y is a negative even integer, which of the following must be true?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
35)Data Sufficiency->How many positive prime numbers are less than the integer n?
(1) 14 < n < 20
(2) 13 < n < 17
36)Data Sufficiency->If x and y are positive integers, is x divisible by y?
1) x is divisible by 5.
2) y is divisible by 5.
37)Data Sufficiency->How many prime factors does \(x^{37}\) have?
(1) The number of prime factors of 16x is one more than the number of prime factors of \(x^4\)
(2) \(2x^{16}\) has 2 prime factors
38)If a positive integer x has 3 prime factors,how many prime factors does x^3 have?
A)27
B)6
C)3
D)1
E)cannot be determined
39)Data Sufficiency->If k is a positive integer, how many unique prime factors does 14k have ?
(1) k^4 is divisible by 100
(2) 50*k has 2 prime factors
40)Data Sufficiency->If x is an integer, is x^3 divisible by 9?
(1) x^2 is divisible by 9.
(2) x^4 is divisible by 9.
41)Data sufficiency->How many different prime numbers are factors of positive integer n?
(1) 4 different prime numbers are factors of 2n
(2) 4 different prime numbers are factors of n^2
42)If x and y are positive integers, are they consecutive?
1) x+y=3
2) x-y=1
43)Data Sufficiency->Is the integer r divisible by 3?
(1) r is the product of 4 consecutive integers.
(2) r < 25
44)Data Sufficiency->Is x^2*y^4 an integer divisible by 9 ?
(1) x is an integer divisible by 3.
(2) xy is an integer divisible by 9.
45)Data Sufficiency->For positive integers x and y, x^2 = 350y. Is y divisible by 28?
(1) x is divisible by 4.
(2) x^2 is divisible by 28.
46)Data Sufficiency-> How many Prime factors does positive integer \(n\) have ?
A)\(n^3\) has 7 prime factors.
B)\(√n\) has 7 prime factors.
47)If x and y represent the number of factors of 90 and 147 respectively.What is the value of x-y?
48)If n is the product of all the integers from 1 to 10 exclusive,how many factors does n have?
49)How many even factors does 21600 have?
A)32
B)42
C)60
D)25
E)52
50)How many positive odd divisors does 540 have?
A)6
B)8
C)12
D)15
E)24
51)How many of the factors of 72 are divisible by 2?
A)4
B)5
C)6
D)8
E)9
52)The number 100^2 has how many factors?
A) 28
B) 25
C) 24
D) 20
E) 16
53)How many factors of 330 are odd numbers greater than 1?
A)8
B)7
C)6
D)5
E)4
54)If \(N=2^7*3^5*5^6*7^8\). How many factors of \(N\) are divisible by 50 but NOT by 100?
A)240
B)345
C)270
D)120
E)None of these
55)Data Sufficiency->How many positive factors does the positive integer x have?
(1) \(x\) is the product of 3 distinct prime numbers.
(2) \(x\) and \(3^7\) have the same number of positive factors.
56)Data Sufficiency->How many factors does \(x\) have, if \(x\) is a positive integer?
(1) \(x = p^n\), where \(p\) is a prime number
(2) \(n^n = n + n\), where \(n\) is a positive integer
57)Data Sufficiency->If 2,3,5 are the only prime factors of a positive integer p,what is the value of p?
A)p>100
B)p has exactly 12 factors including 1 and 12.
58)Data Sufficiency->Is the positive integer P prime?
(1) 61<P<67
(2) P is not divisible by 2.
59)Which of the following statements is/are true?
A)A perfect square always has an odd number of factors.
B)If a given integer has an odd number of factors then that integer must be a perfect square.
60)How many factors does 64 have?
A)8
B)6
C)7
D)6
E)4
61)Data Sufficiency->Is the positive integer \(n\) divisible by \(6\)?
(1) \(\frac{n^2}{180}\) is an integer.
(2) \(\frac{144}{n^2}\) is an integer
62)If a positive integer x has 5 prime factors, how many prime factors does 3x have?
A) 5
B) 6
C) 8
D) 15
E) cannot be determined
63)How many positive distinct prime factors does 5^20 + 5^17 have?
A) One
B) Two
C) Three
D) Four
E) Five
64)How many prime numbers exist between 200 and 220?
(A) None
(B) One
(C) Two
(D) Three
(E) Four
65)Data Sufficiency->If y is a positive odd integer less than 33,does y have a factor p such that 1<p<y?
A)y+2 is divisible by 3.
B)Units digit of y is 7.
66)If x is a positive integer less than 100, for how many values of x is x/6 a prime number?
A)2
B)6
C)8
D)13
E)17
67)Data Sufficiency->If n is an integer between 10 and 99, is n < 80?
(1) The sum of the two digits of n is a prime number.
(2) Each of the two digits of n is a prime number.
68)If n is a positive integer, which of the following statements are correct?
A)All the prime numbers greater than 3 can be represented in the form 4n +1 or 4n + 3.
B)All numbers of the form 4n + 1 and 4n + 3 are prime numbers.
C)All the prime numbers greater than 3 can be represented in the form 6n – 1 or 6n + 1.
D)All numbers of the form 6n + 1 and 6n - 1 are prime numbers.
69)Data Sufficiency->If p is an integer, is p + 4 a prime number less than 50?
(1) p + 1 is a prime number.
(2) p is a prime number.
70)How many odd factors does the number 2100 have?
A)36
B)24
C)12
D)8
E)cannot be determined.
71)Data Sufficiency->If p is an integer greater than 1, is p a prime number?
(1) p is a factor of 13.
(2) p is a factor of 78.
72)If the number 13 completely divides \(x\), and \(x = a^2 * b\), where a and b are distinct prime numbers, which of these numbers must be divisible by 169?
A)\(a^2\)
B)\(b^2\)
C)\(a*b\)
D)\(a^2*b^2\)
E)\(a^3*b\)
73)Data sufficiency->If p is an integer and p>1,is p a prime number?
A)p has p factors.
B)p is a factor of 26.
74)Data Sufficiency->Is the number of distinct prime factors of the positive integer N more than 4?
(1) N is a multiple of 42.
(2) N is a multiple of 98
75)Data Sufficiency->If k is a positive integer, how many different prime factors does k have?
(1) k/30 is an integer
(2) k < 100
76)Data Sufficiency->If k is a positive integer, how many different prime factors does k have?
(1) k/60 is an integer
(2) k < 100
77)Data Sufficiency->If A and B are positive integers, does A have more prime factors than B?
(1) B is the square root of A
(2) A ÷ B is an integer
78)What is the greatest prime factor of \(12^3 – 96\) ?
A)2
B)3
C)7
D)17
E)19
79)What is the greatest prime factor of 4^17 - 2^28?
A)2
B)3
C)5
D)7
E)11
80)What is the total number of distinct prime factors of 28980?
A)2
B)3
C)4
D)5
E)6
81)What is the greatest prime factor of 3^6 - 1 ?
A. 2
B. 3
C. 7
D. 13
E. 17
82)Find the number of factors of 180 that are in the form (4*k + 2), where k is a non-negative integer?
A)1
B)2
C)3
D)4
E)6
83)Data Sufficiency->If A and B are positive integers, does B have more prime factors than A?
(1) B is the square root of A
(2) A ÷ B is an integer
84)If N = 10! then what is the minimum number which should be multiplied with N to make it a perfect square?
A)7
B)14
C)21
D)28
E)36
85)If N = \(2^2*3^3*5^5\), find the total number of odd and even factors of N.
A)24,96
B)15,30
C)24,48
D)15,96
E)15,48
86)Which of the following numbers is a prime number?
A)2343
B)3457
C)4689
D)7731
E)9861
87)The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?
(A) \(10^9\)
(B) \(10^8\)
(C) \(10^7\)
(D) \(10^6\)
(E) \(10^5\)
88)If n is the product of the integers from 1 to 8, inclusive, how many different prime factors greater than 1 does n have?
(A) four
(B) five
(C) six
(D) seven
(E) eight
89)Find the number of factors of 240 that are in the form (4*p + 2), where p is a non-negative integer?
A)1
B)2
C)3
D)4
E)5
90)Data Sufficiency->What is the value of the integer p ?
(1) Each of the integers 2, 3, and 5 is a factor of p.
(2) Each of the integers 2, 5, and 7 is a factor of p.
91)Data Sufficiency->Does the integer k have at least three different positive prime factors?
(1) k/15 is an integer.
(2) k/10 is an integer.
92)If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18
93)If n = 20! + 17, then n is divisible by which of the following?
I. 15
II. 17
III. 19
(A) None
(B) I only
(C) II only
(D) I and II
(E) II and II
94)If 3 < x < 100, for how many values of x is x/3 the square of a prime number?
(A) Two
(B) Three
(C) Four
(D) Five
(E) Nine
95)How many integer values are there for x such that 1 < 3x + 5 < 17?
A)Two
B)Three
C)Four
D)Five
E)Six
96)For any positive integer n, the sum of the first n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 301?
A. 10,100
B. 20,200
C. 22,650
D. 40,200
E. 45,150
97)If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S ?
(A) S/2
(B) S
(C) 2S
(D) 2S + 2
(E) 4S
98)How many prime numbers between 1 and 100 are factors of 7,150 ?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
99)If n is a prime number between 0 and 100, how many positive divisors does n^3 have?
A)1
B)2
C)3
D)4
E)5
100)If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
A)2
B)5
C)6
D)7
E)14
60.
2)Data Sufficiency-> Is the positive integer z divisible by 12?
A)z is divisible by 3.
B)z is not divisible by 2.
B.
3)Data Sufficiency-> Is integer z divisible by 21?
A)z is divisible by 7.
B)z is divisible by 3.
C.
4)Data Sufficiency-> Is integer z divisible by 48?
A)z is divisible by 12.
B)z is divisible by 4.
E.
5)Data Sufficiency-> Is integer z divisible by 12?
A)z is divisible by 8.
B)z is divisible by 6.
C.
6)Data Sufficiency-> Is integer z divisible by 24?
A)z is divisible by 6.
B)z is divisible by 4.
E.
7)How many prime numbers between 1 and 100 are factors of 168?
Three.
8)How many Prime factors does 2180 have?
Three.
2180=2^2*5*109
Notice 109 is a prime number.
Hence, 2180 has three prime factors
2180=2^2*5*109
Notice 109 is a prime number.
Hence, 2180 has three prime factors
9)What is the greatest prime factor of \(49^{19}-7^{35}\)
A)2
B)3
C)7
D)13
E)19
E.
49^19=7^38
Thus => 49^19-7^35=7^38-7^35 => 7^35(7^3-1) = 7^35*342=> 7^35*2*3^2*19
Hence greater Prime factor=19.
49^19=7^38
Thus => 49^19-7^35=7^38-7^35 => 7^35(7^3-1) = 7^35*342=> 7^35*2*3^2*19
Hence greater Prime factor=19.
10)If n is the product of integers from 1 to 12 exclusive,how many prime factors does n have?
Five.
Product = \(2^8*3^4*5^2*7*11\).
Product = \(2^8*3^4*5^2*7*11\).
11)If 2,3,5 are the prime factors of N,what is the value of N?
A)2
B)3
C)5
D)2*3*5
E)cannot be determined.
E.
N=\(2^a*3^b*5^c* P^d\)
Where p is some other prime number.
Basically,There are ∞ numbers with the same set of primes.
N=\(2^a*3^b*5^c* P^d\)
Where p is some other prime number.
Basically,There are ∞ numbers with the same set of primes.
12)If 2,3,5 are the "ONLY" prime factors of K,what is the value of K?
A)2
B)3
C)5
D)2*3*5
E)cannot be determined.
E.
K\(=2^a*3^b*5^c\)
Again -> There are ∞ numbers with the same set of primes.
K\(=2^a*3^b*5^c\)
Again -> There are ∞ numbers with the same set of primes.
13)Data Sufficiency -> What is the value of positive integer K?
A)2,5,7 are prime factors of K.
B)K<100
C.
From statement 1-> K = \(2^a*5^b*7^c*P^z\) ; for any prime number P.
There exist ∞ numbers with the same set of prime numbers.
Hence not sufficient.
From statement 2-> K<100. K can be any integer <100. Clearly not sufficient
Combing them -> Only Value of K possible is 70.
From statement 1-> K = \(2^a*5^b*7^c*P^z\) ; for any prime number P.
There exist ∞ numbers with the same set of prime numbers.
Hence not sufficient.
From statement 2-> K<100. K can be any integer <100. Clearly not sufficient
Combing them -> Only Value of K possible is 70.
14)If p is a positive integer and 200 multiplies by p is square of an integer,what is the value of p?
A)2
B)3
C)5
D)10
E)Cannot be determined.
E.
Notice that the general expression for p will be --> 2 * Prime^even
E.g => p=2
or p=2*5^2
Etc.
Notice that the general expression for p will be --> 2 * Prime^even
E.g => p=2
or p=2*5^2
Etc.
15)Data Sufficiency->What is the value of positive integer p?
A)200 multiplied by p is square of an integer.
B)p is an even number less than 15.
E.
From statement 1-> p=2 or 2*3^2 or 2*2^2 or 2* Any-prime^even--> not sufficient
From statement 2-> p=2 or 4 or any even <15 => Not sufficient
Combining them -> p=2 or p=2*2^2=8
Hence p can be 2 or 8
Thus not sufficient
Hence E.
From statement 1-> p=2 or 2*3^2 or 2*2^2 or 2* Any-prime^even--> not sufficient
From statement 2-> p=2 or 4 or any even <15 => Not sufficient
Combining them -> p=2 or p=2*2^2=8
Hence p can be 2 or 8
Thus not sufficient
Hence E.
16)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is an even number less than 15.
C.
From statement 1-> p=3 or 3*3^2 or 3*7^2 or 3*prime^even--> not sufficient
From statement 2-> p=2 or 4 or any even <15 => Not sufficient
Combining them -> p=3*2^2 =12 is the only possible value.
Hence Sufficient.
Hence C
From statement 1-> p=3 or 3*3^2 or 3*7^2 or 3*prime^even--> not sufficient
From statement 2-> p=2 or 4 or any even <15 => Not sufficient
Combining them -> p=3*2^2 =12 is the only possible value.
Hence Sufficient.
Hence C
17)10^25 – 560 is divisible by all of the following EXCEPT:-
A)11
B)8
C)5
D)4
E)3
E.
Clearly it is divisible by 8,5,4
The real Test here is choosing between 3 and 11.
10^t for any positive integer t will always leave a remainder 1 with 3.
e.g-> 10=> 3*3+1
100=3*33+1
1000=3*333+1
10000=3*3333+1
Hence 10^25=3m+1
Also 560=3k+2 (sum of digits =11,hence remainder with 3-> 2)
Thus 3m+1-3k-2 => 3x-1 for some integer x
Clearly it will never be divisible by 3.
Alternatively,we can also use the "Binomial Expansion".
Clearly it is divisible by 8,5,4
The real Test here is choosing between 3 and 11.
10^t for any positive integer t will always leave a remainder 1 with 3.
e.g-> 10=> 3*3+1
100=3*33+1
1000=3*333+1
10000=3*3333+1
Hence 10^25=3m+1
Also 560=3k+2 (sum of digits =11,hence remainder with 3-> 2)
Thus 3m+1-3k-2 => 3x-1 for some integer x
Clearly it will never be divisible by 3.
Alternatively,we can also use the "Binomial Expansion".
18)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is a factor of 75
E.
Combing the two statements -> p can be 3 or 3*5^2
Combing the two statements -> p can be 3 or 3*5^2
19)Data Sufficiency->What is the value of positive integer p?
A)200 multiplied by p is square of an integer.
B)p is a factor of 60.
C.
Notice that by combing the two statements -> The only value of p possible is 2.
Notice that by combing the two statements -> The only value of p possible is 2.
20)Data sufficiency-> If a and b are prime numbers,what is the value of a*b?
A)a-b=1
B)a=3
A.
Notice that the only consecutive numbers that are prime are -> 2,3
Hence From statement 1-> Product a*b=3*2=6
Notice that the only consecutive numbers that are prime are -> 2,3
Hence From statement 1-> Product a*b=3*2=6
21)If x and y are prime numbers, each greater than 2, which of the following must be true?
I. x+y is an even integer
II. xy is an odd integer
III. (x/y) is not an integer
A. II only
B. I and II only
C. I and III only
D. II and III only
E. I, II, and III
B.
As x and y are prime numbers both greater than 2 => they both must be odd
Hence x+y=odd+odd=even
x*y=odd*odd=odd
Notice that if x=y => x/y will be 1. Hence III is not always true.
If x and y would happen to be different primes then III will be always true.
As x and y are prime numbers both greater than 2 => they both must be odd
Hence x+y=odd+odd=even
x*y=odd*odd=odd
Notice that if x=y => x/y will be 1. Hence III is not always true.
If x and y would happen to be different primes then III will be always true.
22)Which of the following statements must be true->
A)zero is neither prime nor composite.
B)one is neither prime nor composite.
C)-3 and 3 are both primes.
A and B.
Notice -3 is negative and hence cannot be prime
Notice -3 is negative and hence cannot be prime
23)Which of the following statements must be true->
1)A prime number must be positive.
2)For any prime number p,there is no x such that 1<x<p and x is a divisor of p.
3)The product of first ten primes is even.
4)All prime numbers greater than 71 are odd.
5)2 and 3 are the only consecutive integers that are also prime numbers.
6)p is a prime number and x and y are positive integers.If p=x*y then one out of x or y must be one.
7)All the prime numbers greater than 3 can be written as either 4n+1 or 4n-1.
8)All the prime numbers greater than 3 can be written as either 6+1 or 6n-1.
All statements are true.
24)Data Sufficiency->If p is a prime number,what is the value of p?
A)11<p<17
B)p is divisible by 13.
D.
25)Data Sufficiency->If A and B are positive integers, is A/B an integer?
(1) Every factor of B is also a factor of A.
(2) Every prime factor of B is also a prime factor of A.
A.
Statement 1->Using the property => "Every number is a factor of itself "
B must be a factor of B
Hence B must be a factor of A.
Thus A/B must be an integer -> Sufficient
Statement 2-> We can use test cases to Prove this insufficient
Case 1->\(\frac{2*3*7}{2*3*7}\) =1=Integer
Case 2->\(\frac{2*3*7*11}{2^5}\)=> Not an integer.
Hence insufficient
Hence A.
Source-> GMAT-Prep
Statement 1->Using the property => "Every number is a factor of itself "
B must be a factor of B
Hence B must be a factor of A.
Thus A/B must be an integer -> Sufficient
Statement 2-> We can use test cases to Prove this insufficient
Case 1->\(\frac{2*3*7}{2*3*7}\) =1=Integer
Case 2->\(\frac{2*3*7*11}{2^5}\)=> Not an integer.
Hence insufficient
Hence A.
Source-> GMAT-Prep
26)Data Sufficiency->Is \(\frac{n}{14}\) an integer?
(1) \(n\) is divisible by 28.
(2) \(n\) is divisible by 70.
D.
Statement 1->n=28k = 14*2k=14k'
Statement 2->n=70k= 14*5k=14k'
Hence D.
Alternatively,we can use the property => "A number is divisible by factors as well as factors of its factors"
Statement 1->n=28k = 14*2k=14k'
Statement 2->n=70k= 14*5k=14k'
Hence D.
Alternatively,we can use the property => "A number is divisible by factors as well as factors of its factors"
27)If x is a prime number, what is the value of x?
(1) There are a total of 50 prime numbers between 71 and x, inclusive.
(2) There is no integer n such that x is divisible by n and 1 < n < x.
A.
In statement 1->Notice that 71 is 20th prime.
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71.
Hence x will be 50th prime from 71.
Hence sufficient.
Statement 2-> This is just another way of saying that x is prime.
Hence not sufficient
Hence A.
In statement 1->Notice that 71 is 20th prime.
2
3
5
7
11
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71.
Hence x will be 50th prime from 71.
Hence sufficient.
Statement 2-> This is just another way of saying that x is prime.
Hence not sufficient
Hence A.
28)Data Sufficiency->If p = (n)(5^x)(3^k), is p divisible by 10?
(1) n, x, and k are even.
(2) x > n > k > 0
C.
Firstly we are not given in p,x,k,n are an integers or not.
If p is an integer -> Then for p to be divisible by 10 => It must be dividable by 5 and 2=> It must have both 5 and 2 as its primes.
Hence => n must be even and n,k,x must be all pastime integers.
Statement 1->
All n,k,x are even.
Case 1-> \(-2*5^{-10}*3^{-10}\)=>p is not an integer
Case 2->\(2*5^2*3^4\)=> P is an integer and is dividable by 10.
Hence not sufficient.
Statement 2->
All are positive
Again lest use test cases
Case 1->\(1*5*3\)=> p is not divisible by 10
Case 2-> \(2*5*3\)=> p is divisible by 10
Combing the two statements => n must be a positive integer and x,k are positive even numbers.
Hence p must be dividable both 2 and 5.
Hence p must be dividable by 10.
Hence C.
Firstly we are not given in p,x,k,n are an integers or not.
If p is an integer -> Then for p to be divisible by 10 => It must be dividable by 5 and 2=> It must have both 5 and 2 as its primes.
Hence => n must be even and n,k,x must be all pastime integers.
Statement 1->
All n,k,x are even.
Case 1-> \(-2*5^{-10}*3^{-10}\)=>p is not an integer
Case 2->\(2*5^2*3^4\)=> P is an integer and is dividable by 10.
Hence not sufficient.
Statement 2->
All are positive
Again lest use test cases
Case 1->\(1*5*3\)=> p is not divisible by 10
Case 2-> \(2*5*3\)=> p is divisible by 10
Combing the two statements => n must be a positive integer and x,k are positive even numbers.
Hence p must be dividable both 2 and 5.
Hence p must be dividable by 10.
Hence C.
29)Data Sufficiency->If x is a prime number,what is the value of x?
1)There are 3 prime numbers between 11 and x
2 )x+1 is a prime number
B.
30)If \(A\) is positive integer,then \(A\) , \(A^3\)and \(A^7\) will have the exact same prime factors.
A)True
B)False
True.
Property -> \(X\) and \(X^n\) always have the exact same prime factors.
Property -> \(X\) and \(X^n\) always have the exact same prime factors.
31)Data Sufficiency->If y and x are positive integers, is y divisible by 3?
(1) y=x(x+1)
(2) y^2 is divisible by 9.
B.
32)Data Sufficiency->If y and x are positive integers, is y divisible by x?
(1) y = x^2 + x
(2) x has the same prime factors as y.
A.
33)Data Sufficiency->If z is a positive integer,is (z^31+7)^2 divisible by 4?
A)√z has five prime factors.
B)All prime factors of z^3 are greater than 7.
B.
Property in action -> For any number to be odd -> Two must not be its prime.
Also-> \(X\) and \(X^n\) always have the exact same prime factors.
Property in action -> For any number to be odd -> Two must not be its prime.
Also-> \(X\) and \(X^n\) always have the exact same prime factors.
34)If x is a positive odd integer and y is a negative even integer, which of the following must be true?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
C.
35)Data Sufficiency->How many positive prime numbers are less than the integer n?
(1) 14 < n < 20
(2) 13 < n < 17
B.
Remembering the first ten primes can be extremely helpful in these kind of questions.
2
3
5
7
11
13
17
19
23
29.
Remembering the first ten primes can be extremely helpful in these kind of questions.
2
3
5
7
11
13
17
19
23
29.
36)Data Sufficiency->If x and y are positive integers, is x divisible by y?
1) x is divisible by 5.
2) y is divisible by 5.
E.
37)Data Sufficiency->How many prime factors does \(x^{37}\) have?
(1) The number of prime factors of 16x is one more than the number of prime factors of \(x^4\)
(2) \(2x^{16}\) has 2 prime factors
C.This question is based on a very important property -> \(X\) and \(X^n\) always have the exact same prime factors.
So \(x^{37}\) will have the exact same prime factors as x.
Thus,here the question is asking us to get the number of prime factors of \(x\).
Statement 1-> \(16x\) has one more prime factor than \(x^4\)
Now \(x\) and \(x^4\) will have the exact same prime factors.
Hence \(16x\) has one more prime than \(x\)
\(16x=2^4*x\) => Hence 2 cannot be the prime factor of \(x\).
But we still don't have the number of prime factors of \(x\).
Hence not sufficient.
Statement 2-> \(2*x^{16}\) has two prime factors
Here again \(x^{16}\) will have the exact same primes as \(x\).
So as \(2*x^{16}\) has two prime factors => \(x\) can have either one prime factor if 2 isn't one of its prime .
Or two prime factors if 2 is one of its prime.
Hence not sufficient.
Combining the two statements=>
As 2 cannot be one of the primes of x => x must have only one prime factor
Hence sufficient.
Hence C.
So \(x^{37}\) will have the exact same prime factors as x.
Thus,here the question is asking us to get the number of prime factors of \(x\).
Statement 1-> \(16x\) has one more prime factor than \(x^4\)
Now \(x\) and \(x^4\) will have the exact same prime factors.
Hence \(16x\) has one more prime than \(x\)
\(16x=2^4*x\) => Hence 2 cannot be the prime factor of \(x\).
But we still don't have the number of prime factors of \(x\).
Hence not sufficient.
Statement 2-> \(2*x^{16}\) has two prime factors
Here again \(x^{16}\) will have the exact same primes as \(x\).
So as \(2*x^{16}\) has two prime factors => \(x\) can have either one prime factor if 2 isn't one of its prime .
Or two prime factors if 2 is one of its prime.
Hence not sufficient.
Combining the two statements=>
As 2 cannot be one of the primes of x => x must have only one prime factor
Hence sufficient.
Hence C.
38)If a positive integer x has 3 prime factors,how many prime factors does x^3 have?
A)27
B)6
C)3
D)1
E)cannot be determined
C.
\(X\) and \(X^n\) always have the exact same prime factors.
\(X\) and \(X^n\) always have the exact same prime factors.
39)Data Sufficiency->If k is a positive integer, how many unique prime factors does 14k have ?
(1) k^4 is divisible by 100
(2) 50*k has 2 prime factors
C.
\(14k=2*7*k\)
To get the number of prime factors of 14k => we need the number of prime factors of k
Statement 1-> Using the property -> \(X\) and \(X^n\) always have the exact same primes => k must have both 2 and 5 as its primes.
General expression for k= \(2^a*5^b*P^z\)
Where a and b are non negative integers and P is any other prime number and z≥0
Hence not sufficient.
Statement 2->
Form this statement => k can be \(2^a\) or \(5^b\) or \(2^a*5^b\)
Or k=1
Hence not sufficient.
Combing the two statements => k must have only 2 and 5 as its prime factors
Hence 14k will have 3 prime factors
Hence Sufficient
Hence C.
\(14k=2*7*k\)
To get the number of prime factors of 14k => we need the number of prime factors of k
Statement 1-> Using the property -> \(X\) and \(X^n\) always have the exact same primes => k must have both 2 and 5 as its primes.
General expression for k= \(2^a*5^b*P^z\)
Where a and b are non negative integers and P is any other prime number and z≥0
Hence not sufficient.
Statement 2->
Form this statement => k can be \(2^a\) or \(5^b\) or \(2^a*5^b\)
Or k=1
Hence not sufficient.
Combing the two statements => k must have only 2 and 5 as its prime factors
Hence 14k will have 3 prime factors
Hence Sufficient
Hence C.
40)Data Sufficiency->If x is an integer, is x^3 divisible by 9?
(1) x^2 is divisible by 9.
(2) x^4 is divisible by 9.
D.
Property in action -> \(X\) and \(X^n\) always have the exact same prime factors.
Property in action -> \(X\) and \(X^n\) always have the exact same prime factors.
41)Data sufficiency->How many different prime numbers are factors of positive integer n?
(1) 4 different prime numbers are factors of 2n
(2) 4 different prime numbers are factors of n^2
B.
The question is just asking us the prime factors for n.
Statement 1-> From this statement we can conclude that n can have either 4 or 3 prime factors.
Hence Not sufficient
Statement 2-> Using the Property => X and X^n always have the exact same prime factors => As n^2 has 4 prime factors => n myst have 4 prime factors too.
Hence B.
The question is just asking us the prime factors for n.
Statement 1-> From this statement we can conclude that n can have either 4 or 3 prime factors.
Hence Not sufficient
Statement 2-> Using the Property => X and X^n always have the exact same prime factors => As n^2 has 4 prime factors => n myst have 4 prime factors too.
Hence B.
42)If x and y are positive integers, are they consecutive?
1) x+y=3
2) x-y=1
D.
43)Data Sufficiency->Is the integer r divisible by 3?
(1) r is the product of 4 consecutive integers.
(2) r < 25
A.
Property in action -> Product of n consecutive integers is always divisible by n factorial.
So from statement 1-> r=24k for some integer k.
Property in action -> Product of n consecutive integers is always divisible by n factorial.
So from statement 1-> r=24k for some integer k.
44)Data Sufficiency->Is x^2*y^4 an integer divisible by 9 ?
(1) x is an integer divisible by 3.
(2) xy is an integer divisible by 9.
E.
Using test cases can be extremely helpful here.
Clearly statement 1 and 2 are not sufficient on their own.
Combining them lets make test cases =>
Case 1=>
x=3
y=3
x^2*y^4 will be divisible by 9.
Case 2=>
x=18
y=1/2
x^2*y^4 will not be divisible by 9.
Hence E.
Using test cases can be extremely helpful here.
Clearly statement 1 and 2 are not sufficient on their own.
Combining them lets make test cases =>
Case 1=>
x=3
y=3
x^2*y^4 will be divisible by 9.
Case 2=>
x=18
y=1/2
x^2*y^4 will not be divisible by 9.
Hence E.
45)Data Sufficiency->For positive integers x and y, x^2 = 350y. Is y divisible by 28?
(1) x is divisible by 4.
(2) x^2 is divisible by 28.
A.
We are told that x and y are both positive integers.
x^2=350y
As x is an integer => x^2 must be a perfect square.
x^2=350y=2*5^2*7*y
Hence the general expression for y would be -> 2*7*p^z
where p is an integer and z is even.
We are asked if y is divisible by 28 or not.
Clearly y is already divisible by 7 and 2
Hence for y to be divisible by 28=> p must be even too.
Statement 1->
x is divisible by 4
Hence x^2 must have 2^4.
As x=2*5^2*7*y => For x^2 to have a 2^4 => y must have a 2^3.
Hence y must have a 2^3 and it already has a 7.
Thus y must be divisible by 28
Hence sufficient.
Statement2->
Here lets use test cases
y=2*7 =>y will not be divisible by 28
y=2*7*2^2 => y will be divisible by 28.
Hence not sufficient.
Hence A.
We are told that x and y are both positive integers.
x^2=350y
As x is an integer => x^2 must be a perfect square.
x^2=350y=2*5^2*7*y
Hence the general expression for y would be -> 2*7*p^z
where p is an integer and z is even.
We are asked if y is divisible by 28 or not.
Clearly y is already divisible by 7 and 2
Hence for y to be divisible by 28=> p must be even too.
Statement 1->
x is divisible by 4
Hence x^2 must have 2^4.
As x=2*5^2*7*y => For x^2 to have a 2^4 => y must have a 2^3.
Hence y must have a 2^3 and it already has a 7.
Thus y must be divisible by 28
Hence sufficient.
Statement2->
Here lets use test cases
y=2*7 =>y will not be divisible by 28
y=2*7*2^2 => y will be divisible by 28.
Hence not sufficient.
Hence A.
46)Data Sufficiency-> How many Prime factors does positive integer \(n\) have ?
A)\(n^3\) has 7 prime factors.
B)\(√n\) has 7 prime factors.
D.
\(X\) and \(X^n\) always have exact same prime factors.
In other words => "Exponentiation does not change the prime factors of any positive integer X."
\(X\) and \(X^n\) always have exact same prime factors.
In other words => "Exponentiation does not change the prime factors of any positive integer X."
47)If x and y represent the number of factors of 90 and 147 respectively.What is the value of x-y?
6.
\(90=2*3^2*5\)
Number of factors of \(90 =>2*3*2=> 12\)
So x=12
\(147=3*7^2\)
Number of factors of \(147=>2*3=>6\)
So y=6
Hence x-y=12-6=6.
\(90=2*3^2*5\)
Number of factors of \(90 =>2*3*2=> 12\)
So x=12
\(147=3*7^2\)
Number of factors of \(147=>2*3=>6\)
So y=6
Hence x-y=12-6=6.
48)If n is the product of all the integers from 1 to 10 exclusive,how many factors does n have?
160.
49)How many even factors does 21600 have?
A)32
B)42
C)60
D)25
E)52
C.
21600=2^5*3^3*5^2
For even factors -> The power of 2 should at-least be 1.
Hence possible cases -> 5*4*3=>60.
Additionally, for odd factors -> The power of 2 must be zero.
Possible cases -> 1*4*3=> 12.
21600=2^5*3^3*5^2
For even factors -> The power of 2 should at-least be 1.
Hence possible cases -> 5*4*3=>60.
Additionally, for odd factors -> The power of 2 must be zero.
Possible cases -> 1*4*3=> 12.
50)How many positive odd divisors does 540 have?
A)6
B)8
C)12
D)15
E)24
B.
Here 540=2^2*3^3*5
Odd factors => 1*4*2=>8
Additionally Even factors =>2*4*2=>16
Total Factors =>3*4*2=>24.
Here 540=2^2*3^3*5
Odd factors => 1*4*2=>8
Additionally Even factors =>2*4*2=>16
Total Factors =>3*4*2=>24.
51)How many of the factors of 72 are divisible by 2?
A)4
B)5
C)6
D)8
E)9
E.
72=2^3*3^2
For Even divisors ->Power of 2 must be at-least 1.
Possible cases=>3*3=> 9.
Hence E.
72=2^3*3^2
For Even divisors ->Power of 2 must be at-least 1.
Possible cases=>3*3=> 9.
Hence E.
52)The number 100^2 has how many factors?
A) 28
B) 25
C) 24
D) 20
E) 16
B.
53)How many factors of 330 are odd numbers greater than 1?
A)8
B)7
C)6
D)5
E)4
B.
330=2*5*3*11
For Odd factors => Power of 2 must be zero
Possible cases -> 1*2*2*2=>8
So 330 has 8 odd divisors including one.
Excluding one => 330 must have 7 odd divisors.
Hence B.
330=2*5*3*11
For Odd factors => Power of 2 must be zero
Possible cases -> 1*2*2*2=>8
So 330 has 8 odd divisors including one.
Excluding one => 330 must have 7 odd divisors.
Hence B.
54)If \(N=2^7*3^5*5^6*7^8\). How many factors of \(N\) are divisible by 50 but NOT by 100?
A)240
B)345
C)270
D)120
E)None of these
C.
For factors to be divisible by 50 and not divisible by 100 -->
Allowed powers of 2->one
Allowed powers of 5->5
Allowed powers of 3->6
Allowed powers of 7->9
Hence possible cases ->1*5*6*9=>270.
Hence C.
For factors to be divisible by 50 and not divisible by 100 -->
Allowed powers of 2->one
Allowed powers of 5->5
Allowed powers of 3->6
Allowed powers of 7->9
Hence possible cases ->1*5*6*9=>270.
Hence C.
55)Data Sufficiency->How many positive factors does the positive integer x have?
(1) \(x\) is the product of 3 distinct prime numbers.
(2) \(x\) and \(3^7\) have the same number of positive factors.
D.
56)Data Sufficiency->How many factors does \(x\) have, if \(x\) is a positive integer?
(1) \(x = p^n\), where \(p\) is a prime number
(2) \(n^n = n + n\), where \(n\) is a positive integer
C.
From statement 1-> Number of factors of x will be (n+1)
Since we don't have any clue of n => not sufficient.
From statement 2-> n must be 2 as no other value of n will satisfy the equation.
Since we don't have any clue of x => not sufficient.
Combining the two statements => Number of factors of x-> n+1=2+1=3
Hence C.
From statement 1-> Number of factors of x will be (n+1)
Since we don't have any clue of n => not sufficient.
From statement 2-> n must be 2 as no other value of n will satisfy the equation.
Since we don't have any clue of x => not sufficient.
Combining the two statements => Number of factors of x-> n+1=2+1=3
Hence C.
57)Data Sufficiency->If 2,3,5 are the only prime factors of a positive integer p,what is the value of p?
A)p>100
B)p has exactly 12 factors including 1 and 12.
C.
Since 2,3,5 are the only prime factors of p=> The general expression for p will be \(2^a*3^b*5^c\) where a,b,c are positive integers.
Statement 1->p>100.
There exist infinite such cases.
In-fact there are only a few cases where p will be less than 100.
Hence not sufficient.
Statement 2->
Factors =12
Hence (a+1)*(b+1)*(c+1)=12=2*2*3
NOTE-> We can't arrange 12 as 1*1*12 or 2*6*1 or 4*3*1 as a,b,c are positive integers.
Hence a,b,c => (1,1,2) or (1,2,1) or (2,1,1)
Posible values of p=> 150,90,60
Hence not sufficient.
Combining the two statements => p must be 150.
Hence C.
This is probably the hardest Question in this set.
Since 2,3,5 are the only prime factors of p=> The general expression for p will be \(2^a*3^b*5^c\) where a,b,c are positive integers.
Statement 1->p>100.
There exist infinite such cases.
In-fact there are only a few cases where p will be less than 100.
Hence not sufficient.
Statement 2->
Factors =12
Hence (a+1)*(b+1)*(c+1)=12=2*2*3
NOTE-> We can't arrange 12 as 1*1*12 or 2*6*1 or 4*3*1 as a,b,c are positive integers.
Hence a,b,c => (1,1,2) or (1,2,1) or (2,1,1)
Posible values of p=> 150,90,60
Hence not sufficient.
Combining the two statements => p must be 150.
Hence C.
This is probably the hardest Question in this set.
58)Data Sufficiency->Is the positive integer P prime?
(1) 61<P<67
(2) P is not divisible by 2.
A.
There are no prime numbers between 61 and 67.
There are no prime numbers between 61 and 67.
59)Which of the following statements is/are true?
A)A perfect square always has an odd number of factors.
B)If a given integer has an odd number of factors then that integer must be a perfect square.
Both A and B are true.
Two Important Properties ->
A perfect Square always has an odd number of factors
If a number has an odd number of factors => It must be a perfect square.
Two Important Properties ->
A perfect Square always has an odd number of factors
If a number has an odd number of factors => It must be a perfect square.
60)How many factors does 64 have?
A)8
B)6
C)7
D)6
E)4
C.
Method 1=> 64=2^6
Hence it will have 7 factors
Hence C.
Method 2=> 64 is a perfect square =>Thus, It will always have an of Odd number of factors.
The only answer choice that is odd is C.
Method 1=> 64=2^6
Hence it will have 7 factors
Hence C.
Method 2=> 64 is a perfect square =>Thus, It will always have an of Odd number of factors.
The only answer choice that is odd is C.
61)Data Sufficiency->Is the positive integer \(n\) divisible by \(6\)?
(1) \(\frac{n^2}{180}\) is an integer.
(2) \(\frac{144}{n^2}\) is an integer
A.
For n to be divisible by 6 => it must be divisible by both 2 and 3.
Statement 1->
As n^2 is divisible by 180
n^2 must have both 2 and 3 as its prime factors.
But X and X^p always have the exact same prime factors.
Hence n must also have both 2 and 3 as its primes too.
Hence n must be divisible by 6=> Sufficient
Statement 2->
Using test cases ->
n=1=> \(\frac{144}{n^2}\) will be an integer=>NO n is not divisible by 6.
n=6=>\(\frac{144}{n^2}\) will be an integer => YES n is divisible by 6.
Hence not Sufficient.
Hence A.
For n to be divisible by 6 => it must be divisible by both 2 and 3.
Statement 1->
As n^2 is divisible by 180
n^2 must have both 2 and 3 as its prime factors.
But X and X^p always have the exact same prime factors.
Hence n must also have both 2 and 3 as its primes too.
Hence n must be divisible by 6=> Sufficient
Statement 2->
Using test cases ->
n=1=> \(\frac{144}{n^2}\) will be an integer=>NO n is not divisible by 6.
n=6=>\(\frac{144}{n^2}\) will be an integer => YES n is divisible by 6.
Hence not Sufficient.
Hence A.
62)If a positive integer x has 5 prime factors, how many prime factors does 3x have?
A) 5
B) 6
C) 8
D) 15
E) cannot be determined
E.
Let \(x=(p1)^a *(p2)^b *(p3)^c *(p4)^d *(p5)^e\)
Where \(p1,p2,p3,p4,p5\) are prime factors of x and a,b,c,d,e are positive integers.
\(3x=3*(p1)^a *(p2)^b *(p3)^c *(p4)^d *(p5)^e\)
Hence if any one of p1,p2,p3,p4,p5 is 3 => 3x will have 5 prime factors.
Else => 3x will have 6 prime factors.
Hence 3x may have 5 or 6 prime factors.
So there is no unique answer.
Hence E.
Let \(x=(p1)^a *(p2)^b *(p3)^c *(p4)^d *(p5)^e\)
Where \(p1,p2,p3,p4,p5\) are prime factors of x and a,b,c,d,e are positive integers.
\(3x=3*(p1)^a *(p2)^b *(p3)^c *(p4)^d *(p5)^e\)
Hence if any one of p1,p2,p3,p4,p5 is 3 => 3x will have 5 prime factors.
Else => 3x will have 6 prime factors.
Hence 3x may have 5 or 6 prime factors.
So there is no unique answer.
Hence E.
63)How many positive distinct prime factors does 5^20 + 5^17 have?
A) One
B) Two
C) Three
D) Four
E) Five
D.
\(5^{17}+5^{20}=> 5^{17}(5^3+1) = 5^{17}*126=> 2*3^2*5^{17}*7\)
So it has 4 prime factors.
Hence D
\(5^{17}+5^{20}=> 5^{17}(5^3+1) = 5^{17}*126=> 2*3^2*5^{17}*7\)
So it has 4 prime factors.
Hence D
64)How many prime numbers exist between 200 and 220?
(A) None
(B) One
(C) Two
(D) Three
(E) Four
B.
Property->In order check whether a given number is prime or not => We just need to divide it by all the prime numbers less than or equal to the square root of that number.
Highest value in this set-> 220
\(√220\)=14.something
Hence we need to check the divisibility with primes less than 14.something
=> {2,3,5,7,11,13}
All primes >2 are odd
So to shorten our calculation we just need to check the odd numbers.
201-> Divisible by 3
203->Divisible by 7
205->Divisible by 5
207->Divisible by 3
209->Divisible by 11
211-> PRIME=> Not divisible by any prime numbers in our list.
213-> Divisible by 3
215->Divisible by 5
217->Divisible by 7
219->Divisible by 3
Hence only 211 is a prime number.
Hence B.
Property->In order check whether a given number is prime or not => We just need to divide it by all the prime numbers less than or equal to the square root of that number.
Highest value in this set-> 220
\(√220\)=14.something
Hence we need to check the divisibility with primes less than 14.something
=> {2,3,5,7,11,13}
All primes >2 are odd
So to shorten our calculation we just need to check the odd numbers.
201-> Divisible by 3
203->Divisible by 7
205->Divisible by 5
207->Divisible by 3
209->Divisible by 11
211-> PRIME=> Not divisible by any prime numbers in our list.
213-> Divisible by 3
215->Divisible by 5
217->Divisible by 7
219->Divisible by 3
Hence only 211 is a prime number.
Hence B.
65)Data Sufficiency->If y is a positive odd integer less than 33,does y have a factor p such that 1<p<y?
A)y+2 is divisible by 3.
B)Units digit of y is 7.
C.
Note-> Does y have a factor p such that 1<p<y can be rephrased as -> Is y prime?
The Question is just asking us if y prime or not.
Statement 1-> y=> {1,4,7,10,13,16,19,22,25,28,31}
Hence not sufficient.
Statement 2->
y can be ->
7-> Prime
17-> Prime
27-> Composite
Hence not sufficient
Combing the two statements -> y=7 is the only possible value that satisfies both the statements.
Hence y=7
So y is prime
Hence C.
Note-> Does y have a factor p such that 1<p<y can be rephrased as -> Is y prime?
The Question is just asking us if y prime or not.
Statement 1-> y=> {1,4,7,10,13,16,19,22,25,28,31}
Hence not sufficient.
Statement 2->
y can be ->
7-> Prime
17-> Prime
27-> Composite
Hence not sufficient
Combing the two statements -> y=7 is the only possible value that satisfies both the statements.
Hence y=7
So y is prime
Hence C.
66)If x is a positive integer less than 100, for how many values of x is x/6 a prime number?
A)2
B)6
C)8
D)13
E)17
B.
67)Data Sufficiency->If n is an integer between 10 and 99, is n < 80?
(1) The sum of the two digits of n is a prime number.
(2) Each of the two digits of n is a prime number.
B.
We can use test cases to arrive at an answer here.
Statement 1->
n=92=> sum=11 which is prime number.
So n is not less than 80.
n=12=>sum=3 which is a prime number.
So n is less than 80.
Hence not sufficient.
Statement 2->
Both the digits of n are prime.
Allowed tens digit of n->{3,5,7}
So n must be of the form 3X or 5X or 7X where X is the units digit of n.
So,n can never be greater than 80 as both 8 and 9 are non prime numbers.
Hence sufficient.
Hence B.
Source->Gmat-Prep
We can use test cases to arrive at an answer here.
Statement 1->
n=92=> sum=11 which is prime number.
So n is not less than 80.
n=12=>sum=3 which is a prime number.
So n is less than 80.
Hence not sufficient.
Statement 2->
Both the digits of n are prime.
Allowed tens digit of n->{3,5,7}
So n must be of the form 3X or 5X or 7X where X is the units digit of n.
So,n can never be greater than 80 as both 8 and 9 are non prime numbers.
Hence sufficient.
Hence B.
Source->Gmat-Prep
68)If n is a positive integer, which of the following statements are correct?
A)All the prime numbers greater than 3 can be represented in the form 4n +1 or 4n + 3.
B)All numbers of the form 4n + 1 and 4n + 3 are prime numbers.
C)All the prime numbers greater than 3 can be represented in the form 6n – 1 or 6n + 1.
D)All numbers of the form 6n + 1 and 6n - 1 are prime numbers.
A and C are correct.
An important property of prime numbers ->Every prime number greater than 3 can be written as=>
6n+1 OR 6n-1
6n+5 OR 6n-5
4n+1 OR 4n-1
4n+3 OR 4n-3
This can be verified taking the first few primes.
But alternatively every number of these forms isn't prime.
E.g-> 25=6n+1 or 4n+1 and it isn't prime.
An important property of prime numbers ->Every prime number greater than 3 can be written as=>
6n+1 OR 6n-1
6n+5 OR 6n-5
4n+1 OR 4n-1
4n+3 OR 4n-3
This can be verified taking the first few primes.
But alternatively every number of these forms isn't prime.
E.g-> 25=6n+1 or 4n+1 and it isn't prime.
69)Data Sufficiency->If p is an integer, is p + 4 a prime number less than 50?
(1) p + 1 is a prime number.
(2) p is a prime number.
C.
Note=>2 and 3 are the only consecutive integers that are also prime numbers.
Note=>2 and 3 are the only consecutive integers that are also prime numbers.
70)How many odd factors does the number 2100 have?
A)36
B)24
C)12
D)8
E)cannot be determined.
C.
\(2100=2^2*3*5^2*7\)
For odd factors => Power of 2 must be zero.
Possible cases -> \(1*2*3*2=>12\).
Hence C.
\(2100=2^2*3*5^2*7\)
For odd factors => Power of 2 must be zero.
Possible cases -> \(1*2*3*2=>12\).
Hence C.
71)Data Sufficiency->If p is an integer greater than 1, is p a prime number?
(1) p is a factor of 13.
(2) p is a factor of 78.
A.
We are told that p is an integer greater than 1 and we are asked if p is a prime number or not.
Statement 1-> p is a factor of 13.
But 13 has got only 2 factors as it is a prime number.
Factors => 1 and 13
so p must be either 1 or 13.
But p>1
Hence p must be 13.
So p is definitely a prime number.
Sufficient.
Statement 2->
p is factor of 78.
Lets use test cases=>
p=2->2 is a factor of 78 and 2 is a prime number.
p=78->78 is a factor of 78 and clearly 78 isn't a prime number .
Hence not sufficient.
Hence A.
Source->GMAT-prep
We are told that p is an integer greater than 1 and we are asked if p is a prime number or not.
Statement 1-> p is a factor of 13.
But 13 has got only 2 factors as it is a prime number.
Factors => 1 and 13
so p must be either 1 or 13.
But p>1
Hence p must be 13.
So p is definitely a prime number.
Sufficient.
Statement 2->
p is factor of 78.
Lets use test cases=>
p=2->2 is a factor of 78 and 2 is a prime number.
p=78->78 is a factor of 78 and clearly 78 isn't a prime number .
Hence not sufficient.
Hence A.
Source->GMAT-prep
72)If the number 13 completely divides \(x\), and \(x = a^2 * b\), where a and b are distinct prime numbers, which of these numbers must be divisible by 169?
A)\(a^2\)
B)\(b^2\)
C)\(a*b\)
D)\(a^2*b^2\)
E)\(a^3*b\)
D.
As 13 is a prime factor of x => either a or b will be 13.
Hence a^b*b^2 will always be divisible by 169
As 13 is a prime factor of x => either a or b will be 13.
Hence a^b*b^2 will always be divisible by 169
73)Data sufficiency->If p is an integer and p>1,is p a prime number?
A)p has p factors.
B)p is a factor of 26.
A.
There are only two numbers for which the number of factors = number itself.
They are ->{1,2}
As p>1 => p must be 2.
So p=2
Clearly p is a prime number.
Hence Sufficient.
Statement 2->Using test cases
p=2-> 2 is a factor of 26 and 2 is a prime number.
p=26-> 26 is a factor of 26 and 26 is not a prime number.
Clearly not sufficient.
Hence A.
There are only two numbers for which the number of factors = number itself.
They are ->{1,2}
As p>1 => p must be 2.
So p=2
Clearly p is a prime number.
Hence Sufficient.
Statement 2->Using test cases
p=2-> 2 is a factor of 26 and 2 is a prime number.
p=26-> 26 is a factor of 26 and 26 is not a prime number.
Clearly not sufficient.
Hence A.
74)Data Sufficiency->Is the number of distinct prime factors of the positive integer N more than 4?
(1) N is a multiple of 42.
(2) N is a multiple of 98
E.
75)Data Sufficiency->If k is a positive integer, how many different prime factors does k have?
(1) k/30 is an integer
(2) k < 100
C.
76)Data Sufficiency->If k is a positive integer, how many different prime factors does k have?
(1) k/60 is an integer
(2) k < 100
C.
77)Data Sufficiency->If A and B are positive integers, does A have more prime factors than B?
(1) B is the square root of A
(2) A ÷ B is an integer
A.
Property in action->\(X\) and \(X^n\) always have the exact same prime factors.
Property in action->\(X\) and \(X^n\) always have the exact same prime factors.
78)What is the greatest prime factor of \(12^3 – 96\) ?
A)2
B)3
C)7
D)17
E)19
D.
\(12^3-96=2^6*3^3-2^5*3=2^5*3(2*3^2-1)=2^5*3(18-1)=2^5*3*17\)
Clearly here the greatest prime Factor is 17.
\(12^3-96=2^6*3^3-2^5*3=2^5*3(2*3^2-1)=2^5*3(18-1)=2^5*3*17\)
Clearly here the greatest prime Factor is 17.
79)What is the greatest prime factor of 4^17 - 2^28?
A)2
B)3
C)5
D)7
E)11
D.
80)What is the total number of distinct prime factors of 28980?
A)2
B)3
C)4
D)5
E)6
D.
\(28980=2^2*3^2*5*7*23\)
Hence it has 5 prime factors.
Hence D.
\(28980=2^2*3^2*5*7*23\)
Hence it has 5 prime factors.
Hence D.
81)What is the greatest prime factor of 3^6 - 1 ?
A. 2
B. 3
C. 7
D. 13
E. 17
D.
using the identity a^2-b^2 = (a+b)(a-b)=> (3^3)^2 -1^2 = 28*26 = 2^3*7*13
Clearly the greatest prime factor will be 13.
using the identity a^2-b^2 = (a+b)(a-b)=> (3^3)^2 -1^2 = 28*26 = 2^3*7*13
Clearly the greatest prime factor will be 13.
82)Find the number of factors of 180 that are in the form (4*k + 2), where k is a non-negative integer?
A)1
B)2
C)3
D)4
E)6
E.
Method 1->
180 is a relatively smaller number=>
Factors can be seen as follows ->
1*180
2*90
3*60
4*45
5*36
6*30
9*20
10*18
12*15
15*12(The repetition has begun.
Hence checking the factors we get => There are 6 factors of the form 4k+2 for k≥0
Method 2->
180=2^2^3^2*5
Factors of the form -> 4k+2 => 2(2k+1)=> 2*odd number.
2*1
2*3
2*3^2
2*5
2*3*5
2*3^2*5
Hence six values.
E.
Method 1->
180 is a relatively smaller number=>
Factors can be seen as follows ->
1*180
2*90
3*60
4*45
5*36
6*30
9*20
10*18
12*15
15*12(The repetition has begun.
Hence checking the factors we get => There are 6 factors of the form 4k+2 for k≥0
Method 2->
180=2^2^3^2*5
Factors of the form -> 4k+2 => 2(2k+1)=> 2*odd number.
2*1
2*3
2*3^2
2*5
2*3*5
2*3^2*5
Hence six values.
E.
83)Data Sufficiency->If A and B are positive integers, does B have more prime factors than A?
(1) B is the square root of A
(2) A ÷ B is an integer
D.
X and X^n have the exact same primes.
So Statement 1 is sufficient.
As for statement 2-> If A/B is an integer => Every prime factor of B must be a prime factor of A.
Hence B can never have more prime factors than A.
Hence Sufficient.
Hence D.
X and X^n have the exact same primes.
So Statement 1 is sufficient.
As for statement 2-> If A/B is an integer => Every prime factor of B must be a prime factor of A.
Hence B can never have more prime factors than A.
Hence Sufficient.
Hence D.
84)If N = 10! then what is the minimum number which should be multiplied with N to make it a perfect square?
A)7
B)14
C)21
D)28
E)36
A.
10!=1*2*3*4*5*6*7*8*9*10=> 2^8*3^4*5^2*7
In a perfect square the Power of every prime must be even.
Thus =>Smallest number to be multiplied will be 7.
Hence A.
10!=1*2*3*4*5*6*7*8*9*10=> 2^8*3^4*5^2*7
In a perfect square the Power of every prime must be even.
Thus =>Smallest number to be multiplied will be 7.
Hence A.
85)If N = \(2^2*3^3*5^5\), find the total number of odd and even factors of N.
A)24,96
B)15,30
C)24,48
D)15,96
E)15,48
C.
For even factors => Power of 2 must be at-least 1.
Hence possible cases => 2*4*6 =>48
odd factors => Power of 2 must be zero.
Possible cases =>1*4*6=>24
Hence 48,24 is the correct option.
For even factors => Power of 2 must be at-least 1.
Hence possible cases => 2*4*6 =>48
odd factors => Power of 2 must be zero.
Possible cases =>1*4*6=>24
Hence 48,24 is the correct option.
86)Which of the following numbers is a prime number?
A)2343
B)3457
C)4689
D)7731
E)9861
B.
Every number except the second one is dividable by 3 so they can never be prime.
Hence B must be a prime number.
Alternatively prime numbers can be written as either 6n+1 OR 6n-1
Only B can be written as 6n+1
One of the other options can be written as 6n+1 or 6n-1.
Hence B is a prime number.
Every number except the second one is dividable by 3 so they can never be prime.
Hence B must be a prime number.
Alternatively prime numbers can be written as either 6n+1 OR 6n-1
Only B can be written as 6n+1
One of the other options can be written as 6n+1 or 6n-1.
Hence B is a prime number.
87)The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?
(A) \(10^9\)
(B) \(10^8\)
(C) \(10^7\)
(D) \(10^6\)
(E) \(10^5\)
Product=>2*3*5*7*11*13*17*19=> 10*20*150*300=>9000000=> Close to 1 crore or 10^7 then it is close to 10 lakhs or 10^6
Hence C.
Source->Official Guide.
Hence C.
Source->Official Guide.
88)If n is the product of the integers from 1 to 8, inclusive, how many different prime factors greater than 1 does n have?
(A) four
(B) five
(C) six
(D) seven
(E) eight
A.
Source->Official Guide.
Source->Official Guide.
89)Find the number of factors of 240 that are in the form (4*p + 2), where p is a non-negative integer?
A)1
B)2
C)3
D)4
E)5
D.
90)Data Sufficiency->What is the value of the integer p ?
(1) Each of the integers 2, 3, and 5 is a factor of p.
(2) Each of the integers 2, 5, and 7 is a factor of p.
E.
Basically n can be any integer that has 2,3,5,7 as its prime factors.
Eg-> 2*3*5*7
OR
2*3*5*7*13
Etc.
Hence E.
Source->Official Guide.
Basically n can be any integer that has 2,3,5,7 as its prime factors.
Eg-> 2*3*5*7
OR
2*3*5*7*13
Etc.
Hence E.
Source->Official Guide.
91)Data Sufficiency->Does the integer k have at least three different positive prime factors?
(1) k/15 is an integer.
(2) k/10 is an integer.
C.
Statement 1->3 and 5 must be the prime factors of K.
Statement 2-> 2 and 5 must be the prime factors of K.
combing them -> 2,3 and 5 must be the prime factors of K.
Hence C.
Source->Official Guide.
Statement 1->3 and 5 must be the prime factors of K.
Statement 2-> 2 and 5 must be the prime factors of K.
combing them -> 2,3 and 5 must be the prime factors of K.
Hence C.
Source->Official Guide.
92)If p is the product of the integers from 1 to 30, inclusive, what is the greatest integer k for which 3^k is a factor of p?
A. 10
B. 12
C. 14
D. 16
E. 18
C.
Source->Official Guide.
Source->Official Guide.
93)If n = 20! + 17, then n is divisible by which of the following?
I. 15
II. 17
III. 19
(A) None
(B) I only
(C) II only
(D) I and II
(E) II and II
C.
Source->Official Guide.
Source->Official Guide.
94)If 3 < x < 100, for how many values of x is x/3 the square of a prime number?
(A) Two
(B) Three
(C) Four
(D) Five
(E) Nine
B.
Source->Official Guide.
Source->Official Guide.
95)How many integer values are there for x such that 1 < 3x + 5 < 17?
A)Two
B)Three
C)Four
D)Five
E)Six
D.
x->(-1.3,4)
So possible integer values for x =>-1,0,1,2,3
Thus,five values for x are possible.
Hence D.
x->(-1.3,4)
So possible integer values for x =>-1,0,1,2,3
Thus,five values for x are possible.
Hence D.
96)For any positive integer n, the sum of the first n positive integers equals n(n+1)/2. What is the sum of all the even integers between 99 and 301?
A. 10,100
B. 20,200
C. 22,650
D. 40,200
E. 45,150
B.
Source->Oficial Guide
Source->Oficial Guide
97)If the sum of the first n positive integers is S, what is the sum of the first n positive even integers, in terms of S ?
(A) S/2
(B) S
(C) 2S
(D) 2S + 2
(E) 4S
C.
The easiest way here to pick options.
Let n=1
Sum=1
Sum of evens = 2
Hence for S=1=> Sum of evens = 2
Only Option C fits that case.
NOTE-> Sometimes while using test cases more than one value may satisfy.In that case we must use some additional test cases to arrive at the correct answer.
Source->2011 Official Guide.
The easiest way here to pick options.
Let n=1
Sum=1
Sum of evens = 2
Hence for S=1=> Sum of evens = 2
Only Option C fits that case.
NOTE-> Sometimes while using test cases more than one value may satisfy.In that case we must use some additional test cases to arrive at the correct answer.
Source->2011 Official Guide.
98)How many prime numbers between 1 and 100 are factors of 7,150 ?
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
D.
The question is asking us in a fancy way to get the prime factors of 7150 between 1 and 100.
Breaking down 7150=> 2*5^2*11*13
Hence It has 4 prime factors.
Hence D.
Source-> Official Guide.
The question is asking us in a fancy way to get the prime factors of 7150 between 1 and 100.
Breaking down 7150=> 2*5^2*11*13
Hence It has 4 prime factors.
Hence D.
Source-> Official Guide.
99)If n is a prime number between 0 and 100, how many positive divisors does n^3 have?
A)1
B)2
C)3
D)4
E)5
D.
100)If y is the smallest positive integer such that 3,150 multiplied by y is the square of an integer, then y must be
A)2
B)5
C)6
D)7
E)14
E.
Note-> For any perfect square -> Power/exponent of each prime factor must be even.
Source-> Official Guide
Note-> For any perfect square -> Power/exponent of each prime factor must be even.
Source-> Official Guide
Note-->PM me if you find any answers/solutions inaccurate/inadequate.
Any Feedback would be Appreciated.
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Day-3
Mock Test 3
Number of Questions->85
Topic Covered-> Evens/odds
Source-> This Quiz contains questions from varied sources along with quite a few self-made questions.
1)Which of the following statements must be true?
A)Any integer can be either even or odd but not both.
B)2n is always even.
C)2n+1 is always odd.
D)Zero is an even integer.
E)The sum of even number of odd numbers is always is always even.
F)The sum of odd number of odd numbers is always odd.
G)If any factor of an integer is even,then the integer itself would be even.
2)Data Sufficiency->If n is a positive integer,is n divisible by 2?
A)7n-8 is divisible by 20.
B)3n^2+2n+5 is a prime number.
3)If x,y,z are integers and p=x*y*z,if p odd?
A)x is odd.
B)z is even.
4)If x and n are non negative integers,then which of the following statements must be true?
A)If x is even then x^n will always be even.
B)If x is odd then x^n will always be odd.
5)Data Sufficiency->If \(A\) and \(B\) are integers,is \(A*B^4\)even?
A)\((2A+B)^3\) is even.
B)\(A+3B\) is odd.
6)If w,x,y,z are consecutive positive integers then which of the following statements will always be odd?
I) wx+(y+2)^2*z
II) w^x+z^y
III) 2w^3-3x^2-5y+6z^2
A)I
B)II
C)III
D)I and II
E)II and III
7)Which of the following CANNOT be the sum of two prime numbers?
(A) 19
(B) 45
(C) 68
(D) 79
(E) 88
8)Data Sufficiency->If \(x\) and \(y\) are integers, is \(3*x^4 + 4*y\) even?
(1) \(x^3\) is even.
(2) \(y^{2x} + 3\) is even.
9)If x and y are integers is y odd?
A)y^2x +8 is odd.
B)y^7 is odd.
10)Data Sufficiency->If \(X\) is a positive integer, is \(X^2 + 1\) an odd number?
(1) \(X\) is the smallest integer that is divisible by all integers from 11 to 15, inclusive.
(2) \(3^X\) is an odd number.
11)Data Sufficiency-> If a, b, and k are positive integers, is the sum (a + b) an even number or an odd number?
(1) a = ( k^3 + 3k^2 + 3k + 6)
(2) b = (k^2 + 4a +5)
12)Data Sufficiency->If x, y, and z are positive integers, where x is an odd number and z = x^2 + y^2 + 4. Is y^2 divisible by 4?
(1) Z = 8k -3 where k is a positive integer.
(2) When (z-x+1) is divided by 2, it leaves a remainder.
13)If a, b, c, d, and e are integers and the expression \(\frac{a*b^2*c^2}{d^2*e}\) gives a positive even integer, which of the following options must be true?
I. \(a*b*c\) is even
II. \(\frac{a}{e}\) is positive
III. \(\frac{a}{d^2}\) is positive.
A)I
B)II
C)I and II
D)I and III
E)I,II,III
14)Data Sufficiency->If y is a positive integer is y divisible by 21?
A)y^2 is divisible by 21.
B)y^3 is divisible by 63.
15)If r and s are positive integers, and \(r^2\) + \(\frac{r}{s}\) is an odd integer, which of the following cannot be even?
A)\(3*r+2*s\)
B)\((r-1)*(s+2)\)
C)\(r^{s+1}+s^r\)
D)\(r^3+3\)
E)\(s^4+4\)
16)Data Sufficiency->If x and y are integers, is y even?
(1) (x + 2) * (y^2 + 7) is even.
(2) (x^3 + 8) * (y^2 -4) is even.
17)Data Sufficiency->If x and y are integers, is x even?
(1) (x + 2) * (y^2 + 7) is even.
(2) (x^3 + 8) * (y^2 -4) is even.
18)If A is a positive integer, then which of the following statements is true?
1. A^2 + A -1 is always even.
2. (A^4+1)(A^4+2) + 3A is even only when A is even.
3. (A-1)(A+2)(A+4) is never odd.
A)1
B)2
C)3
D)1 and 2
E)2 and 3
19)Data Sufficiency-->If the positive integer N divisible by 2?
A)11n+8 is divisible by 4.
B)5n^2+2n+13 is a prime number.
20)Set S is given as S = {1,3,5,7,9,11,13,15,17}. In how many ways can three numbers be chosen from Set S such that the sum of those three numbers is 18?
A)zero
B)two
C)three
D)six
E)nine
21)Data Sufficiency ->If A and B are positive integers,is the Product A*B even?
A)A is odd
B)B^43 is even
22)Which of the following statements must be true?
The product of first 100 prime numbers is even.
The sum of first 100 prime numbers is odd.
The sum of first five non-negative even numbers is divisible by both 4 and 5.
A)I only
B)II only
C)I and II only
D)I and III only
E)I, II and III
23)If p,q,r,s are consecutive positive integers,then which of the following statements must be odd ?
I)p*s + [(q+14)^2] *r
II)p^r + s^q
III)26*p^3 + 3*q^23 + 5*r^21 + 28*s^2
A)I only
B)II only
C)I and II only
D)II and III only
E)I, II and III
24)Data Sufficiency->If x, y, z and w are positive integers, is x odd?
(1) 7x + 8y + 4z + 5w is odd.
(2) 3x + 2y + 8z + 2w is even.
25)7)Data Sufficiency->Is z even?
(1) \(\frac{z}{2}\) is even.
(2) 3z is even.
26)Data Sufficiency->Is z even?
(1) 5z is even.
(2) 3z is even.
27)Data Sufficiency->If \(x\) and \(y\) are positive integers and \(x\) is odd, is \(x*y\) even?
(1) \(x^3*y = 6*a^3 + 23\) where a is a positive integer.
(2) \(x^2+y = 3*k + 7\) where \(k\) is a positive integer.
28)Data Sufficiency -> Is z even?
(1) 5z is even.
(2) 15z is even.
29)Data Sufficiency->If a and b are integers, is b even?
(1) 3a + 4b is even.
(2) 3a + 5b is even.
30)If n is an integer, then which of the following statements is/are FALSE?
I)\(n^3 – n\) is always even.
II)\(8n^3 +12n^2 +6n +1\) is always even.
III)\(√ (4n^2 – 4*n +1)\) is always odd.
A)I only
B)II only
C)I and II only
D)II and III only
E)I, II and III
31)Data Sufficiency ->If t is a positive integer, is t^3 + 1 an odd number?
(1) t is the smallest integer that is divisible by all integers from 21 to 25, inclusive.
(2) 5^t is an odd number.
32)Data Sufficiency ->If x and y are integers, is 31x^4 + 4y even?
(1) \(x^{13}\) is even.
(2) \(y^{12x}\) + 3 is even.
33)Data Sufficiency ->If p, q, and r are positive integers, where p is an odd number and r = p^2 + q^3 + 4. Is q^3 divisible by 8?
(1) r = 18k -5 where k is a positive integer
(2) When (r-p+13) is divided by 2, it leaves a remainder.
34)Data Sufficiency ->If a, b, and t are positive integers, is the sum (a + b) an even number or an odd number?
(1) a = ( t^13 + t^12 + 3t+ 6)
(2) b = (t^2 + 4a +5)
35)Data Sufficiency ->If x any y are non negative integers, is y^2 divisible by 4?
A)x>1
B)y^2x is odd.
36)Data Sufficiency ->If x any y are non negative integers, is y^2 divisible by 4?
A)x>1
B)y^2x is even.
37)Data Sufficiency->Is p^2 divisible by 4?
A)p^2 is divisible by 2.
B)p=2n, where n is a positive number.
38)Data Sufficiency->If p is an integer,Is p^2 divisible by 4?
A)p^2 is divisible by 2.
B)p=2n, where is n is a positive integer.
39)If a and b are positive integers, and a^3 + a^2/b is an odd integer, which of the following cannot be even?
A)31a + 12b
B(a-1)(s+10)
C)ab+21b
D)a^3 +7
E)b^4 +4
40)Set S is given as S = {1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37}. In how many ways can three numbers be chosen from Set S such that the sum of those three numbers is 68?
A)zero
B)1
C)19
D)38
E)19!
41)Data Sufficiency->If a+b is even, is b an integer?
1) a-b is even.
2) a+3b is even.
42)Which of the following statements must be true?
I)The product of first 249 prime numbers is even.
II)The sum of first 249 prime numbers is even.
III)Sum of any 13 prime numbers greater than 100 is always odd.
A)I only
B)II only
C)I and II only
D)I and III only
E)I, II and III
43)State True/False->The sum of first 5 non negative of 5 is divisible by both 25 and 50.
A)True
B)False
44)Data Sufficiency-> If z is an integer, is z even?
(1) z/2 is not an odd integer.
(2) z + 5 is an odd integer.
45)Data Sufficiency->Is the integer a even?
(1) a is divisible by 7.
(2) a is divisible by 9.
46)Data Sufficiency-> Is a even?
(1) 2a is even.
(2) √a is even.
47)Data Sufficiency->Is n an even?
1) 3n is even.
2) 7n is even.
48)Data Sufficiency-> If n is an integer, is n even?
(1) 2n is divisible by 4.
(2) n^2 is even.
49)If n is an integer, which of the following must be even?
(A) n+1
(B) n+2
(C) 2n
(D) 2n +1
(E) n^2
50)Data Sufficiency->\(w\), \(x\), \(y\) and \(z\) are all integers. Is \(w*x*y*z\) even?
(1) \(w*x*y\) is odd.
(2) \(x*y – z\) is even.
51)Data Sufficiency->If x, y, and z are all positive integers, is x + y + z even?
1) \(\frac{2x}{(y+z)}\) = odd.
2) \(\frac{x^2}{4yz}\) is an integer.
52)Data Sufficiency->If x is an integer, is x even?
(1) ax is even, where a is an integer.
(2) x^2 - 1 is an odd integer.
53)Data Sufficiency->If x is an integer, is (x + p)(x + q) an even integer?
(1) q is an even integer.
(2) p is an even integer.
54)Data Sufficiency->If x is an integer, is (x + p)(x + q) an even integer?
(1) q is an even integer.
(2) p is an odd integer.
55)Data Sufficiency->If x is a positive integer,is x even?
(1)x^2+y^2=98
(2)x is odd.
56)Data Sufficiency->If x any y are positive integers,is x even?
(1)x^2+y^2=98
(2)x=y.
57)Data Sufficiency->If x, y and z are integers and xy + z is an even integer, is x an even integer?
(1) xy + xz is an even integer.
(2) y + xz is an odd integer.
58)If both x and y are positive even integers, then which of the following expressions must also
be even?
I. \(y^{x − 1}\)
II. \(y – 1\)
III. \(\frac{x}{2}\)
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
59)If both x and y are positive integers and y is even,then which of the following expressions must also
be even?
I. \(y^{x − 1}\)
II. \(x^{y-1}\)
III. \(\frac{[fraction]y^2}{2}[/fraction]\)
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
60)If a is an odd integer and b is an even integer, which of the following CANNOT be true?
A. a + b is an odd integer.
B. a is a factor of b.
C. b is a factor of a.
D. a^b is an odd integer.
E. b^a is an even integer.
61)If x is a positive odd integer and y is a negative even integer, which of the following must be true?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
62)If x is a positive odd integer and y is a negative even integer, which of the following must be false?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
63)If x is an even integer and y is an odd integer, which of the following must be an even integer?
A. y/x
B. x+y
C. 3x + 2y
D. 3(x + y)
E. 2y/x
64)If positive integers x and y are not both odd, which of the following must be even?
(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1
65)If positive integers x and y are not both even, which of the following must be even?
A)xy
B)x+y
C)x-y^2
D)xy(x+y)
E)x/y
66)If positive integers x and y are not both odd, which of the following can never be even?
(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1
67)If x and y are integers and x + y = 5, which of the following must be true?
A) x and y are consecutive integers.
B) If x < 0, then y > 0.
C) If x > 0, then y < 0.
D) Both x and y are even.
E) Both x and y are less than 5.
68)If a and b are odd integers, which of the following must be an even integer?
A. a(b – 2)
B. ab + 4
C. (a + 2)(b – 4)
D. 3a + 5b
E. a(a + 6)
69)If p is an even integer and q is an odd integer, which of the following must be an odd integer?
A. p/q
B. pq
C. 2p+q
D. 2(p+q)
E. 3p/q
70)If a and b are even integers, which of the following is an odd integer?
A. ab + 2
B. a(b − 1)
C. a(a + 5)
D. 3a + 4b
E. (a + 3)(b − 1
71)If a is an even integer and b is an odd integer which of the following must be an odd integer :
A. a/b
B. ab
C. 2a+b
D. 2(a+b)
E. 3a/b
72)If x and y are integers and x^2*y is a negative odd integer, which of the following must be true?
I. xy^2 is odd.
II. xy is negative.
III. x + y is even.
A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III
73)If x is an odd integer and y and z are even integers, which of the following CANNOT be an integer?
A. y/z
B. x/y
C. z/x
D. yx/z
E. zx/y
74)X and Y are integers, X is even and negative, Y is odd and positive. Which of the following could be false?
1. (X + Y) is an odd number.
2. Y^(X + Y) is an integer.
3. X^Y is a positive number.
A. 2 only.
B. 3 only.
C. 1 and 3 only.
D. 2 and 3 only.
E. 1, 2 and 3.
75)If x is an odd negative integer and y is an even integer, which of the following statements must be true?
I. (3x - 2y) is odd
II. xy^2 is an even negative integer
III. (y^2 - x) is an odd negative integer
A. I only
B. II only
C. I and II
D. I and III
E. II and III
76)If x, y, and z are positive integers and x^2 = y^2 + z^2, which of the following must be true?
I. x > z
II. x = y + z
III. y^2 + z^2 is a positive integer
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
77)If x and y are positive and \(x^2+y^2\)=100, then for which of the following is the value of x+y greatest?
A)x=10
B)x=9
C)x=8
D)x=7
E)x=6
78)If x is an even integer, which of the following is an odd integer?
(A) 3x + 2
(B) 7x
(C) 8x +5
(D) x^2
(E) x^3
eq
79)If x, y, and z are consecutive even positive integers, which of the following could be equal to x + y + z ?
A)141
B)200
C)318
D)391
E)412
80)If x, y, and z are consecutive odd positive integers, which of the following could be equal to x + y + z ?
A)9
B)221
C)400
D)900
E)1200
81)If w,x,y are consecutive odd positive integers with w<x<y, then which of the following maybe the value of y-x-w?
A)-4
B)-2
C)-1
D)0
E)3
82)If a is an odd integer, which of the following must be an even integer?
A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)^2
E. None of the above.
83)If a and b are odd integers, which of the following must be an even integer?
A. a^2(b – 2)
B. ab + 40
C. (a + 22)(b – 42)
D. 31a + 51b
E. a(a + 16)
84)If a and b are odd integers which of the following must be an even integer?
A. a/b
B. ab
C. (ab)^2
D. ab + b
E. (a + b)/b
85)If x is even integer, which of the following must be an odd integer?
A. \(\frac{3x}{2}\)
B. \(\frac{3x}{2} + 1\)
C. \(3x^2\)
D. \(\frac{3x^2}{2}\)
E. \(\frac{3x^2}{2} + 1\)
A)Any integer can be either even or odd but not both.
B)2n is always even.
C)2n+1 is always odd.
D)Zero is an even integer.
E)The sum of even number of odd numbers is always is always even.
F)The sum of odd number of odd numbers is always odd.
G)If any factor of an integer is even,then the integer itself would be even.
A,D,E,F,G
Note -> Since n can be an integer or a non integer -> 2n may be even or odd.
Similarly 2n+1 may be even or odd.
Note -> Since n can be an integer or a non integer -> 2n may be even or odd.
Similarly 2n+1 may be even or odd.
2)Data Sufficiency->If n is a positive integer,is n divisible by 2?
A)7n-8 is divisible by 20.
B)3n^2+2n+5 is a prime number.
D.
3)If x,y,z are integers and p=x*y*z,if p odd?
A)x is odd.
B)z is even.
B.
4)If x and n are non negative integers,then which of the following statements must be true?
A)If x is even then x^n will always be even.
B)If x is odd then x^n will always be odd.
B.
Notice that n can be zero too.
Although x and x^n will always have the same even/odd nature,we must make sure than the exponent is non zero.
Anything^zero=1 which is always odd.
Notice that n can be zero too.
Although x and x^n will always have the same even/odd nature,we must make sure than the exponent is non zero.
Anything^zero=1 which is always odd.
5)Data Sufficiency->If \(A\) and \(B\) are integers,is \(A*B^4\)even?
A)\((2A+B)^3\) is even.
B)\(A+3B\) is odd.
D.
6)If w,x,y,z are consecutive positive integers then which of the following statements will always be odd?
I) wx+(y+2)^2*z
II) w^x+z^y
III) 2w^3-3x^2-5y+6z^2
A)I
B)II
C)III
D)I and II
E)II and III
E.
7)Which of the following CANNOT be the sum of two prime numbers?
(A) 19
(B) 45
(C) 68
(D) 79
(E) 88
D.
Essentially this is testing our knowledge on Even/odd numbers.
If sum of 2 primes is even => 2 must not be either of them.
If sum of 2 primes is odd => 2 must be one of them.
Hence as 79 =2+77 =>79 can never be written as sum of 2 prime numbers.
Hence D.
Essentially this is testing our knowledge on Even/odd numbers.
If sum of 2 primes is even => 2 must not be either of them.
If sum of 2 primes is odd => 2 must be one of them.
Hence as 79 =2+77 =>79 can never be written as sum of 2 prime numbers.
Hence D.
8)Data Sufficiency->If \(x\) and \(y\) are integers, is \(3*x^4 + 4*y\) even?
(1) \(x^3\) is even.
(2) \(y^{2x} + 3\) is even.
A.
9)If x and y are integers is y odd?
A)y^2x +8 is odd.
B)y^7 is odd.
B.
Note-> Statement 1 is not sufficient as we gotta consider the case of x=0.
Note-> Statement 1 is not sufficient as we gotta consider the case of x=0.
10)Data Sufficiency->If \(X\) is a positive integer, is \(X^2 + 1\) an odd number?
(1) \(X\) is the smallest integer that is divisible by all integers from 11 to 15, inclusive.
(2) \(3^X\) is an odd number.
A.
11)Data Sufficiency-> If a, b, and k are positive integers, is the sum (a + b) an even number or an odd number?
(1) a = ( k^3 + 3k^2 + 3k + 6)
(2) b = (k^2 + 4a +5)
C.
12)Data Sufficiency->If x, y, and z are positive integers, where x is an odd number and z = x^2 + y^2 + 4. Is y^2 divisible by 4?
(1) Z = 8k -3 where k is a positive integer.
(2) When (z-x+1) is divided by 2, it leaves a remainder.
D.
13)If a, b, c, d, and e are integers and the expression \(\frac{a*b^2*c^2}{d^2*e}\) gives a positive even integer, which of the following options must be true?
I. \(a*b*c\) is even
II. \(\frac{a}{e}\) is positive
III. \(\frac{a}{d^2}\) is positive.
A)I
B)II
C)I and II
D)I and III
E)I,II,III
C.
14)Data Sufficiency->If y is a positive integer is y divisible by 21?
A)y^2 is divisible by 21.
B)y^3 is divisible by 63.
D.
15)If r and s are positive integers, and \(r^2\) + \(\frac{r}{s}\) is an odd integer, which of the following cannot be even?
A)\(3*r+2*s\)
B)\((r-1)*(s+2)\)
C)\(r^{s+1}+s^r\)
D)\(r^3+3\)
E)\(s^4+4\)
D.
16)Data Sufficiency->If x and y are integers, is y even?
(1) (x + 2) * (y^2 + 7) is even.
(2) (x^3 + 8) * (y^2 -4) is even.
E.
17)Data Sufficiency->If x and y are integers, is x even?
(1) (x + 2) * (y^2 + 7) is even.
(2) (x^3 + 8) * (y^2 -4) is even.
C.
18)If A is a positive integer, then which of the following statements is true?
1. A^2 + A -1 is always even.
2. (A^4+1)(A^4+2) + 3A is even only when A is even.
3. (A-1)(A+2)(A+4) is never odd.
A)1
B)2
C)3
D)1 and 2
E)2 and 3
E.
19)Data Sufficiency-->If the positive integer N divisible by 2?
A)11n+8 is divisible by 4.
B)5n^2+2n+13 is a prime number.
D.
20)Set S is given as S = {1,3,5,7,9,11,13,15,17}. In how many ways can three numbers be chosen from Set S such that the sum of those three numbers is 18?
A)zero
B)two
C)three
D)six
E)nine
A.
Notice that all the integers in the set are odd
So whenever we pick any 3 numbers => Sum will be odd+odd+odd=> odd
Hence sum will never be 18 which is even.
Hence A.
Notice that all the integers in the set are odd
So whenever we pick any 3 numbers => Sum will be odd+odd+odd=> odd
Hence sum will never be 18 which is even.
Hence A.
21)Data Sufficiency ->If A and B are positive integers,is the Product A*B even?
A)A is odd
B)B^43 is even
B.
22)Which of the following statements must be true?
The product of first 100 prime numbers is even.
The sum of first 100 prime numbers is odd.
The sum of first five non-negative even numbers is divisible by both 4 and 5.
A)I only
B)II only
C)I and II only
D)I and III only
E)I, II and III
E.
Properties in action => Sum of odd number of odd numbers -> Odd
Sum of even number of odd numbers is even.
Two is the only even prime.
Properties in action => Sum of odd number of odd numbers -> Odd
Sum of even number of odd numbers is even.
Two is the only even prime.
23)If p,q,r,s are consecutive positive integers,then which of the following statements must be odd ?
I)p*s + [(q+14)^2] *r
II)p^r + s^q
III)26*p^3 + 3*q^23 + 5*r^21 + 28*s^2
A)I only
B)II only
C)I and II only
D)II and III only
E)I, II and III
D.
24)Data Sufficiency->If x, y, z and w are positive integers, is x odd?
(1) 7x + 8y + 4z + 5w is odd.
(2) 3x + 2y + 8z + 2w is even.
B.
25)7)Data Sufficiency->Is z even?
(1) \(\frac{z}{2}\) is even.
(2) 3z is even.
A.
Notice in statement 2 z may or may not be an integer.
Notice in statement 2 z may or may not be an integer.
26)Data Sufficiency->Is z even?
(1) 5z is even.
(2) 3z is even.
C.
27)Data Sufficiency->If \(x\) and \(y\) are positive integers and \(x\) is odd, is \(x*y\) even?
(1) \(x^3*y = 6*a^3 + 23\) where a is a positive integer.
(2) \(x^2+y = 3*k + 7\) where \(k\) is a positive integer.
A.
28)Data Sufficiency -> Is z even?
(1) 5z is even.
(2) 15z is even.
E.
29)Data Sufficiency->If a and b are integers, is b even?
(1) 3a + 4b is even.
(2) 3a + 5b is even.
C.
30)If n is an integer, then which of the following statements is/are FALSE?
I)\(n^3 – n\) is always even.
II)\(8n^3 +12n^2 +6n +1\) is always even.
III)\(√ (4n^2 – 4*n +1)\) is always odd.
A)I only
B)II only
C)I and II only
D)II and III only
E)I, II and III
B.
Notice that for statement 3-> 4n^2-4n+1 is (2n-1)^2
Notice that for statement 3-> 4n^2-4n+1 is (2n-1)^2
31)Data Sufficiency ->If t is a positive integer, is t^3 + 1 an odd number?
(1) t is the smallest integer that is divisible by all integers from 21 to 25, inclusive.
(2) 5^t is an odd number.
A.
32)Data Sufficiency ->If x and y are integers, is 31x^4 + 4y even?
(1) \(x^{13}\) is even.
(2) \(y^{12x}\) + 3 is even.
A.
Also notice that in statement 2,its not a necessity that y will be odd.If x=0 then y can be even or odd.This fact is useless in this questions as we don't want the even/odd nature of y,rather we are only interested in the even/odd nature of x.
Also notice that in statement 2,its not a necessity that y will be odd.If x=0 then y can be even or odd.This fact is useless in this questions as we don't want the even/odd nature of y,rather we are only interested in the even/odd nature of x.
33)Data Sufficiency ->If p, q, and r are positive integers, where p is an odd number and r = p^2 + q^3 + 4. Is q^3 divisible by 8?
(1) r = 18k -5 where k is a positive integer
(2) When (r-p+13) is divided by 2, it leaves a remainder.
D.
Notice that this question is just asking us for the even/odd nature of q.
If q is even => q^3 must be divisible by 8.
Notice that this question is just asking us for the even/odd nature of q.
If q is even => q^3 must be divisible by 8.
34)Data Sufficiency ->If a, b, and t are positive integers, is the sum (a + b) an even number or an odd number?
(1) a = ( t^13 + t^12 + 3t+ 6)
(2) b = (t^2 + 4a +5)
C.
This is a typical question in which the even/odd nature of two integers is connected via a third integer.
This is a typical question in which the even/odd nature of two integers is connected via a third integer.
35)Data Sufficiency ->If x any y are non negative integers, is y^2 divisible by 4?
A)x>1
B)y^2x is odd.
C.
NOTE-> B is alone not sufficient as x can be zero too.
And anything ^0 is one which is always odd.
NOTE-> B is alone not sufficient as x can be zero too.
And anything ^0 is one which is always odd.
36)Data Sufficiency ->If x any y are non negative integers, is y^2 divisible by 4?
A)x>1
B)y^2x is even.
B.
Notice that if y^2x is even => x can never be zero => y must always be even.
Notice that if y^2x is even => x can never be zero => y must always be even.
37)Data Sufficiency->Is p^2 divisible by 4?
A)p^2 is divisible by 2.
B)p=2n, where n is a positive number.
E.
Notice we are not told if p is an integer or not.
E.g=> p=√2 for n=√2/2 and clearly p^2 is not divisible by 4.
Notice we are not told if p is an integer or not.
E.g=> p=√2 for n=√2/2 and clearly p^2 is not divisible by 4.
38)Data Sufficiency->If p is an integer,Is p^2 divisible by 4?
A)p^2 is divisible by 2.
B)p=2n, where is n is a positive integer.
D.
39)If a and b are positive integers, and a^3 + a^2/b is an odd integer, which of the following cannot be even?
A)31a + 12b
B(a-1)(s+10)
C)ab+21b
D)a^3 +7
E)b^4 +4
D.
Notice that if a^3 + a^2/b is odd -> a must be even and b must be even too.
Notice that if a^3 + a^2/b is odd -> a must be even and b must be even too.
40)Set S is given as S = {1,3,5,7,9,11,13,15,17,19,21,23,25,27,29,31,33,35,37}. In how many ways can three numbers be chosen from Set S such that the sum of those three numbers is 68?
A)zero
B)1
C)19
D)38
E)19!
A.
Notice that they are all odd.
Hence, sum of any three integers picked from this set will be odd + odd +odd -> odd
Hence sum can never be 68.
Hence A.
Notice that they are all odd.
Hence, sum of any three integers picked from this set will be odd + odd +odd -> odd
Hence sum can never be 68.
Hence A.
41)Data Sufficiency->If a+b is even, is b an integer?
1) a-b is even.
2) a+3b is even.
D.
Hint => If 2x is even => x must be an integer.
Hint => If 2x is even => x must be an integer.
42)Which of the following statements must be true?
I)The product of first 249 prime numbers is even.
II)The sum of first 249 prime numbers is even.
III)Sum of any 13 prime numbers greater than 100 is always odd.
A)I only
B)II only
C)I and II only
D)I and III only
E)I, II and III
E.
Two properties are in action here.
ONE-> Sum of even number of odd numbers is even and the Sum of odd number of odd numbers is odd.
TWO-> All primes numbers greater than 2 are odd.
Two properties are in action here.
ONE-> Sum of even number of odd numbers is even and the Sum of odd number of odd numbers is odd.
TWO-> All primes numbers greater than 2 are odd.
43)State True/False->The sum of first 5 non negative of 5 is divisible by both 25 and 50.
A)True
B)False
True.
Sum=0+5+10+15+20=50.
50 is divisible by both 25 and 50.
Sum=0+5+10+15+20=50.
50 is divisible by both 25 and 50.
44)Data Sufficiency-> If z is an integer, is z even?
(1) z/2 is not an odd integer.
(2) z + 5 is an odd integer.
B.
We need the even odd nature of integer z
Statement 1->
z=5
z=4
Taking these two test cases we can say that this is an insufficient statement
Statement 2->
z-odd=odd => z=> odd+odd=even
Hence sufficient
Hence B.
We need the even odd nature of integer z
Statement 1->
z=5
z=4
Taking these two test cases we can say that this is an insufficient statement
Statement 2->
z-odd=odd => z=> odd+odd=even
Hence sufficient
Hence B.
45)Data Sufficiency->Is the integer a even?
(1) a is divisible by 7.
(2) a is divisible by 9.
E.
46)Data Sufficiency-> Is a even?
(1) 2a is even.
(2) √a is even.
B.
47)Data Sufficiency->Is n an even?
1) 3n is even.
2) 7n is even.
C.
Note that it is not given in the original question that n is an integer.
Note that it is not given in the original question that n is an integer.
48)Data Sufficiency-> If n is an integer, is n even?
(1) 2n is divisible by 4.
(2) n^2 is even.
D.
Source-> GMAT-prep
Source-> GMAT-prep
49)If n is an integer, which of the following must be even?
(A) n+1
(B) n+2
(C) 2n
(D) 2n +1
(E) n^2
C.
Source->Official Guide.
Source->Official Guide.
50)Data Sufficiency->\(w\), \(x\), \(y\) and \(z\) are all integers. Is \(w*x*y*z\) even?
(1) \(w*x*y\) is odd.
(2) \(x*y – z\) is even.
C.
51)Data Sufficiency->If x, y, and z are all positive integers, is x + y + z even?
1) \(\frac{2x}{(y+z)}\) = odd.
2) \(\frac{x^2}{4yz}\) is an integer.
C.
52)Data Sufficiency->If x is an integer, is x even?
(1) ax is even, where a is an integer.
(2) x^2 - 1 is an odd integer.
B.
53)Data Sufficiency->If x is an integer, is (x + p)(x + q) an even integer?
(1) q is an even integer.
(2) p is an even integer.
E.
54)Data Sufficiency->If x is an integer, is (x + p)(x + q) an even integer?
(1) q is an even integer.
(2) p is an odd integer.
C.
55)Data Sufficiency->If x is a positive integer,is x even?
(1)x^2+y^2=98
(2)x is odd.
C.
56)Data Sufficiency->If x any y are positive integers,is x even?
(1)x^2+y^2=98
(2)x=y.
A.
The only value of a and b that will satisfy statement 1 is 7.
Hence a=b=7
The only value of a and b that will satisfy statement 1 is 7.
Hence a=b=7
57)Data Sufficiency->If x, y and z are integers and xy + z is an even integer, is x an even integer?
(1) xy + xz is an even integer.
(2) y + xz is an odd integer.
B.
58)If both x and y are positive even integers, then which of the following expressions must also
be even?
I. \(y^{x − 1}\)
II. \(y – 1\)
III. \(\frac{x}{2}\)
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
A.
59)If both x and y are positive integers and y is even,then which of the following expressions must also
be even?
I. \(y^{x − 1}\)
II. \(x^{y-1}\)
III. \(\frac{[fraction]y^2}{2}[/fraction]\)
(A) I only
(B) II only
(C) III only
(D) I and III only
(E) I, II, and III
C.
60)If a is an odd integer and b is an even integer, which of the following CANNOT be true?
A. a + b is an odd integer.
B. a is a factor of b.
C. b is a factor of a.
D. a^b is an odd integer.
E. b^a is an even integer.
C.
An even number can never divide an odd number.
Hence C.
An even number can never divide an odd number.
Hence C.
61)If x is a positive odd integer and y is a negative even integer, which of the following must be true?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
C.
62)If x is a positive odd integer and y is a negative even integer, which of the following must be false?
A. x^3 + y is a positive odd integer
B. x^2 + y^2 is a negative odd integer
C. x^0 + y^11 is a negative odd integer
D. x + y is a positive odd integer
E. x + y is a negative odd integer
B.
63)If x is an even integer and y is an odd integer, which of the following must be an even integer?
A. y/x
B. x+y
C. 3x + 2y
D. 3(x + y)
E. 2y/x
C.
64)If positive integers x and y are not both odd, which of the following must be even?
(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1
A.
Source->Official Guide
Source->Official Guide
65)If positive integers x and y are not both even, which of the following must be even?
A)xy
B)x+y
C)x-y^2
D)xy(x+y)
E)x/y
D.
66)If positive integers x and y are not both odd, which of the following can never be even?
(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1
E.
67)If x and y are integers and x + y = 5, which of the following must be true?
A) x and y are consecutive integers.
B) If x < 0, then y > 0.
C) If x > 0, then y < 0.
D) Both x and y are even.
E) Both x and y are less than 5.
B.
We need to check as to which of the statement is always true.
So if we can find a case that makes a statement false we can react the statement.
Statement 1->
-1,6 => Rejected.
Statment 2->
Always true.
Statement 3->
2,3=> Rejected
Statement 4->
2,3 => Rejected.
Statement 5->-1,6 => Rejected.
Hence B.
We need to check as to which of the statement is always true.
So if we can find a case that makes a statement false we can react the statement.
Statement 1->
-1,6 => Rejected.
Statment 2->
Always true.
Statement 3->
2,3=> Rejected
Statement 4->
2,3 => Rejected.
Statement 5->-1,6 => Rejected.
Hence B.
68)If a and b are odd integers, which of the following must be an even integer?
A. a(b – 2)
B. ab + 4
C. (a + 2)(b – 4)
D. 3a + 5b
E. a(a + 6)
D.
69)If p is an even integer and q is an odd integer, which of the following must be an odd integer?
A. p/q
B. pq
C. 2p+q
D. 2(p+q)
E. 3p/q
C.
70)If a and b are even integers, which of the following is an odd integer?
A. ab + 2
B. a(b − 1)
C. a(a + 5)
D. 3a + 4b
E. (a + 3)(b − 1
E.
71)If a is an even integer and b is an odd integer which of the following must be an odd integer :
A. a/b
B. ab
C. 2a+b
D. 2(a+b)
E. 3a/b
C.
72)If x and y are integers and x^2*y is a negative odd integer, which of the following must be true?
I. xy^2 is odd.
II. xy is negative.
III. x + y is even.
A. I only
B. III only
C. I and II only
D. I and III only
E. I, II, and III
D.
We are told that x^2*y<0
Hence x≠0 and y≠0
So as x^2 is always greater than 0.
Hence y<0.
Also as x^2*y is odd => x and y must be both odd.
Option 1->xy^2=> odd*odd^2=> odd.
True.
Option 2-> xy will be negative for x being positive and xy will be positive for x being negative.
Hence this statement is not always true.
Option 3->
x+y=> odd+odd=> even.
Hence this is True.
So the correct answer is D.
We are told that x^2*y<0
Hence x≠0 and y≠0
So as x^2 is always greater than 0.
Hence y<0.
Also as x^2*y is odd => x and y must be both odd.
Option 1->xy^2=> odd*odd^2=> odd.
True.
Option 2-> xy will be negative for x being positive and xy will be positive for x being negative.
Hence this statement is not always true.
Option 3->
x+y=> odd+odd=> even.
Hence this is True.
So the correct answer is D.
73)If x is an odd integer and y and z are even integers, which of the following CANNOT be an integer?
A. y/z
B. x/y
C. z/x
D. yx/z
E. zx/y
B.
An odd number can never be divisible by even number.
Hence B.
An odd number can never be divisible by even number.
Hence B.
74)X and Y are integers, X is even and negative, Y is odd and positive. Which of the following could be false?
1. (X + Y) is an odd number.
2. Y^(X + Y) is an integer.
3. X^Y is a positive number.
A. 2 only.
B. 3 only.
C. 1 and 3 only.
D. 2 and 3 only.
E. 1, 2 and 3.
D.
75)If x is an odd negative integer and y is an even integer, which of the following statements must be true?
I. (3x - 2y) is odd
II. xy^2 is an even negative integer
III. (y^2 - x) is an odd negative integer
A. I only
B. II only
C. I and II
D. I and III
E. II and III
A.
76)If x, y, and z are positive integers and x^2 = y^2 + z^2, which of the following must be true?
I. x > z
II. x = y + z
III. y^2 + z^2 is a positive integer
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
E.
77)If x and y are positive and \(x^2+y^2\)=100, then for which of the following is the value of x+y greatest?
A)x=10
B)x=9
C)x=8
D)x=7
E)x=6
D.
We can use estimation to check the values of x+y in each option.
Option 1-> x=10 => y=0 --> Not allowed as y>0
Option 2-> x=9=> y=4.something => x+y=> 13.something
Option 3-> x=8=> y=6 =>x+y=14
Option 4-> x=7 => y=7.something => x+y=14.something
Option 5=> x=6=>y=8=> x+y=14
Hence clearly Option 4 will give us the greatest value of x+y
Hence D.
We can use estimation to check the values of x+y in each option.
Option 1-> x=10 => y=0 --> Not allowed as y>0
Option 2-> x=9=> y=4.something => x+y=> 13.something
Option 3-> x=8=> y=6 =>x+y=14
Option 4-> x=7 => y=7.something => x+y=14.something
Option 5=> x=6=>y=8=> x+y=14
Hence clearly Option 4 will give us the greatest value of x+y
Hence D.
78)If x is an even integer, which of the following is an odd integer?
(A) 3x + 2
(B) 7x
(C) 8x +5
(D) x^2
(E) x^3
C.
79)If x, y, and z are consecutive even positive integers, which of the following could be equal to x + y + z ?
A)141
B)200
C)318
D)391
E)412
C.
Let the 3 consecutive even integers be represents by -->
2n
2n+2
2n+4
Sum=> 6n+6-> 6(n+1)
Hence the sum must be a multiple of 6.
Only option that fits this criteria is C.
Hence C.
Let the 3 consecutive even integers be represents by -->
2n
2n+2
2n+4
Sum=> 6n+6-> 6(n+1)
Hence the sum must be a multiple of 6.
Only option that fits this criteria is C.
Hence C.
80)If x, y, and z are consecutive odd positive integers, which of the following could be equal to x + y + z ?
A)9
B)221
C)400
D)900
E)1200
A.
81)If w,x,y are consecutive odd positive integers with w<x<y, then which of the following maybe the value of y-x-w?
A)-4
B)-2
C)-1
D)0
E)3
C.
Source->GMAT-prep
Source->GMAT-prep
82)If a is an odd integer, which of the following must be an even integer?
A. a^4−a+1
B. (a^4−a)(a+1/a)
C. a^4−a^3+a^2+2a
D. (a^3+a^2+a)^2
E. None of the above.
B.
Firstly -->Positive exponents does not affect the even/odd nature of any number
Option A-> a^4−a+1=> odd-odd+odd=> even+odd=> odd=> Rejected.
Option B->(a^4−a)(a+1/a)=> (a^3-1)*(a^2+a)=> (odd-odd)*(odd+odd) => even*even => even=> Acceptable.
Option C. a^4−a^3+a^2+2a=> odd-odd+odd+even=>even+odd=>odd=> Rejected.
Option D. (a^3+a^2+a)^2=>(odd+odd+odd)^2=>(even+odd)^2=>odd^2=> odd=>Rejected.
Option E. None of the above.=> Rejected.
Firstly -->Positive exponents does not affect the even/odd nature of any number
Option A-> a^4−a+1=> odd-odd+odd=> even+odd=> odd=> Rejected.
Option B->(a^4−a)(a+1/a)=> (a^3-1)*(a^2+a)=> (odd-odd)*(odd+odd) => even*even => even=> Acceptable.
Option C. a^4−a^3+a^2+2a=> odd-odd+odd+even=>even+odd=>odd=> Rejected.
Option D. (a^3+a^2+a)^2=>(odd+odd+odd)^2=>(even+odd)^2=>odd^2=> odd=>Rejected.
Option E. None of the above.=> Rejected.
83)If a and b are odd integers, which of the following must be an even integer?
A. a^2(b – 2)
B. ab + 40
C. (a + 22)(b – 42)
D. 31a + 51b
E. a(a + 16)
D.
84)If a and b are odd integers which of the following must be an even integer?
A. a/b
B. ab
C. (ab)^2
D. ab + b
E. (a + b)/b
D.
85)If x is even integer, which of the following must be an odd integer?
A. \(\frac{3x}{2}\)
B. \(\frac{3x}{2} + 1\)
C. \(3x^2\)
D. \(\frac{3x^2}{2}\)
E. \(\frac{3x^2}{2} + 1\)
E.
Source->GMAT-Prep.
Source->GMAT-Prep.
22 Nov 2016, 02:31
Kudos
Bookmarks
Day-2
MOCK TEST-2
Advanced Quiz on Divisibility and Prime numbers.
Number of Questions-> 100.
Sources-> The following questions are from various sources and a few are self made.
101)If 2940 multiplied by k is a square of an integer,what is the smallest possible value of k?
A)3
B)6
C)15
D)21
E)cannot be determined.
102)If n is a positive integer and the product of all integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?
A)10
B)11
C)12
D)13
E)14
103)n is a positive integer, and k is the product of all integers from 1 to n inclusive. If k is a multiple of 1440, then the smallest possible value of n is
A)8
B)12
C)6
D)18
E)24
104)If x^2 is divisible by 216, what is the smallest possible value for positive integer x?
105)If k is a positive even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?
A.3
B.9
C.12
D.15
E.20
106)If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?
(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight
107)If N = \(2^2 * 3^3 * 5^5\),How many factors of N are divisible by 5 but not divisible by 3.
A)10
B)12
C)14
D)15
E)20
108)What is the greatest prime factor of 12!11! + 11!10!?
(A) 7
(B) 11
(C) 13
(D) 17
(E) 19
109)Data Sufficiency->How many distinct factors does positive integer k have?
(1) k has more distinct factors than the integer 9 but fewer distinct factors than the integer 81.
(2) k is the product of two distinct prime numbers.
110) What is the greatest prime factor of (2^29) - (2^26)?
A)2
B)4
C)7
D)8
E)3
111)Data Sufficiency->How many different prime factors does positive integer n have?
(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors
112)What is the greatest prime factor of 9919?
A) 7
B) 13
C) 17
D) 97
E) 109
113)If 5x^2 has two different prime factors, at most how many different prime factors does x have?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
114)What is the greatest prime factor of 12!11! + 11!10!?
(A) 7
(B) 11
(C) 13
(D) 17
(E) 19
115)Data Sufficiency->How many different prime numbers are factors of positive integer n?
(1) 4 different prime numbers are factors of \(2n\).
(2) 4 different prime numbers are factors of \(n^2\).
116)What is the greatest prime factor of 3^6 - 1 ?
A. 2
B. 3
C. 7
D. 13
E. 17
117)Data Sufficiency->If m and n are different positive integers, then how many prime numbers are in set {m, n, m + n}?
(1) mn is prime.
(2) m + n is even.
118)What is the greatest prime factor of 1+2+3+。。。+40?
A. 17
B. 29
C. 31
D. 37
E. 41
119)Data Sufficiency->How many positive factors does positive integer \(N\) have?
1) \(N^2\) has three positive factors.
2) \(2N\) has four positive factors.
120)If \(n=3*4*p\) where p is a prime number greater than 3,how many different positive non-prime divisors does n have, excluding 1 and n?
(A) Six
(B) Seven
(C) Eight
(D) Nine
(E) Ten
121)Data Sufficiency->How many divisors does the positive integer \(N\) have?
(1)\(27N^3\) has 16 factors.
(2)\(90<N^3<200\)
122)Data Sufficiency-> How many prime factors does positive integer n have?
(1) n/5 has only a prime factor.
(2) 3*n^2 has two different prime factors.
123)What is the greatest prime factor of \(2^{10}*5^4 - 2^{13}*5^2 + 2^{14}\)?
(A) 2
(B) 3
(C) 7
(D) 11
(E) 13
124)Data Sufficiency->If p is a positive integer, is 2p + 1 prime?
(1) p is prime.
(2) units digit of p is not prime.
125)What is the greatest prime factor of \(6^8−3^8\) ?
A) 3
B) 11
C) 17
D) 19
E) 31
126)Data Sufficiency->If \(p\) is a positive integer, is \(p\) a prime number?
(1) \(p\) and \(p+1\) have the same number of factors.
(2) \(p-1\) is a factor of \(p\).
127)Data Sufficiency->If p is a positive integer, is integer k prime?
(1) 3p + 3 = k
(2) 7! + 3 = k
128)Data Sufficiency->k is a positive integer. Is k prime?
(1) At least one number in the set {1, k, k + 7} is prime.
(2) k is odd.
129)Data Sufficiency->What is the value of the integer p?
(1) p is a prime number.
(2) 88 ≤ p ≤ 95
130)Data Sufficiency->If n is a positive integer, does n have four or more distinct factors?
(1) n is not prime
(2)900 ≤ n < 1100
131)If x is an integer and x^2 is even, which of the following must be true?
I. x is odd.
II. x is even.
III. x^3 is odd.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
132)Data Sufficiency->If K is a positive 3-digit number, is K prime?
(1) The last digit of K is not even
(2) K is the smallest number possible where the hundreds digit is the sum of the tens and units digit. Tens and units digits are equal.
133)Data Sufficiency->If x and y are positive integers, is y divisible by 3?
(1) y = 2x^3 + 9x^2 - 5x.
(2) x is an odd number
134)If \(x\) and \(y\) are positive integer and \(xy\) is divisible by 4, which of the following must be true?
A) If \(x\) is even then \(y\) is odd.
B) If \(x\) is odd then \(y\) is a multiple of 4.
C) If \(x+y\) is odd then \(y/x\) is not an integer.
D) If \(x+y\) is even then \(x/y\) is an integer.
E) \(x^y\) is even.
135)If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
136)Data Sufficiency->If x is a prime number, what is the value of x?
(1) 2x + 2 is the cube of a positive integer.
(2) The average of any x consecutive integers is an integer.
137)The sum of prime numbers that are greater than 60 but less than 70 is
(A) 67
(B) 128
(C) 191
(D) 197
(E) 260
138)Data Sufficiency->If x, y, and z are positive integers, what is the greatest prime factor of the product xyz?
(1) The greatest common factor of x, y, and z is 7.
(2) The lowest common multiple of x, y, and z is 84.
139)If n=6p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n?
A)2
B)3
C)4
D)6
E)cannot be determined.
140)Given that N=a^3*b^4*c^5 where a, b and c are distinct prime numbers, what is the smallest number with which N should be multiplied such that it becomes a perfect square, a perfect cube as well as a perfect fifth power?
A. a^3*b^4*c^5
B. a^5*b^4*c^3
C. a^2*b^3*c^5
D. a^7*b^6*c^5
E. a^27*b^26*c^25
141)Data Sufficiency->Is the integer b divisible by 6 ?
(1) 8b is divisible by 3.
(2) 9b is divisible by 12.
142)Data Sufficiency->If k is a positive integer, is k a prime number?
(1) No integers between 2 and \(\sqrt{k}\), inclusive divides k evenly.
(2) No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.
143)The "prime sum" of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions. For example , the prime sum of 12 is 7, since 12 = 2 x 2 x 3 and 2 +2 + 3 = 7. For which of the following integers is the prime sum greater than 35 ?
(A) 440
(b) 512
(C) 620
(D) 700
(E) 750
144)Data Sufficiency->If a and b are integers, is a + b divisible by 15?
(1) a + b is divisible by 30.
(2) a + b is divisible by 5
145)Data Sufficiency->If A and B are positive integers, is B divisible by A?
(1) 2B/A is an integer.
(2) B^2/A is an integer
146)Data Sufficiency->If n is a positive integer is n-1 divisible by 3 ?
(1) n^2+n is not divisible by 6.
(2) 3n=3k+3 where k is a positive multiple of 3
147)Data Sufficiency->If n is a positive integer, is n – 1 divisible by 3?
(1) n^2 + n is not divisible by 6.
(2) 3n=k+3 where k is a positive multiple of 3.
148)Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set {x + 2, x + 3, x + 4, …, x + 59}?
A)0
B)17
C)18
D)23
E)24
149)The smallest prime factor of 899 is \(x\). Which of the following is true of \(x\)?
A. \(1 \lt x \le 7\)
B. \(7 \lt x \le 14\)
C. \(14 \lt x \le 21\)
D. \(21 \lt x \le 28\)
E. \(28 \lt x \le 35\)
150)Data Sufficiency->If n is a positive integer, does n have four or more distinct factors?
(1) n is not prime
(2) 150 ≤ n < 200
151)Data Sufficiency->If p is a positive integer, what is the value of p?
(1) p/4 is a prime number.
(2) p is divisible by 3
152)Data Sufficiency->If a and b are integers, is (ab+2)(ab+3)(ab+4) divisible by 12?
(1) a=even
(2) b=odd
153)An integer n that is greater than 1 is said to be "prime-saturated" if it has no prime factor greater than or equal to √n.
Which of the following integers is prime saturated?
A) 6
B) 35
C) 46
D) 66
E) 75
154)Data Sufficiency->If p is a positive integer, is p^2 divisible by 96?
(1) p is a multiple of 8.
(2) p^2 is a multiple of 12
155)If x is a positive integer greater than 1, what is the sum of the multiples of x from x to x^2, inclusive?
(A) x(x + 1)(x-1)
(B) x^2(x + 1)/2
(C) x^2(x-1)
(D) (x^3 + 2x)/2
(E) x(x-1)^2
156)Data Sufficiency->Does p^2 = q if p is a prime number?
(1) q^2 – p^2 =0
(2) p^2 = 49
157)A number is said to be prime saturated if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two digit prime saturated integer ?
A. 99
B. 98
C. 97
D. 96
E. 95
158)Data Sufficiency->How many different prime factors does x have?
1) \(5x^2\) has two different prime factors
2) \(x > 1\)
159)How many different prime factors does positive integer x have?
(1) \(1 < x < 6\)
(2) \(5x^2\) has four factors.
160)Data Sufficiency->If N is a positive integer, does N have exactly three factors?
(1) The integer N^2 has exactly five factors
(2) Only one factor of N is a prime number
161)Data Sufficiency->What is the value of x?
(1) x is the square of an integer.
(2) 577 < x < 675
162)Data Sufficiency->If x is a prime number, what is the value of x.
1)There are 4 prime numbers between 11 and x
2 )There is no y such that 1<y<x and y is the divisor of x.
163)Data Sufficiency->If x is a prime number, what is the value of x.
1)There are 3 prime numbers between 11 and x
2 )x+1 is a prime number
164)For how many positive integers is the number of positive divisors equal to the number itself ?
A) none
B) one
C) two
D)three
E)cannot be determined
165) The positive integer 2000 has how many factors?
A)2
B)10
C)12
D)16
E)20
166)Data Sufficiency->If x is a perfect square greater than 1, what is the value of x?
(1) x has exactly 3 distinct factors.
(2) x has exactly one positive odd factor
167)Data Sufficiency->How many factors does y have?
(1) y is the cube of an integer.
(2) y is the product of 2 distinct positive digits.
168)Data Sufficiency->If x, y and k are integers, is xy divisible by 3?
(1) y = 2^(16) - 1
(2) The sum of the digits of x equals 6^k
169)Data Sufficiency->If x is a prime number, what is the value of x?
(1) x is less than 15
(2) (x-2) is multiple of 5
170)Data Sufficiency->How many prime factors does positive integer n have?
(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.
171)If x = 13y, where y is a prime number greater than 2, how many different positive even divisors does x have, including x?
A. 0
B. 1
C. 2
D. 3
E. It cannot be determined from the information given
172)Data Sufficiency ->If b is a positive integer, then b has how many distinct positive factors?
(1) all of b's factors are also factors of 62.
(2) b is the product of one even prime number and one odd prime number.
173)Data Sufficiency->How many different factors does the integer n have?
(1) n = (a^4)(b^3) where a and b are different positive prime numbers.
(2) The only positive prime numbers that are factors of n are 5 and 7.
174)Data Sufficiency->If x and y are positive integers and x + y = 3^x, is y divisible by 6?
(1) x is odd.
(2) x is a multiple of 3.
175)Data Sufficiency ->If x^3 – x = n and x is a positive integer greater than 1, is n divisible by 8?
(1) When 3x is divided by 2, there is a remainder.
(2) x = 4y + 1, where y is an integer.
176)If a positive odd integer N has p positive factors, how many positive factors will 2N have ?
A) p
B) 2p
C) P+1
D) 2p+1
E) Cannot be determined
177)Data Sufficiency->Is the integer n odd?
(1) n is divisible by 3
(2) 2n is divisible by twice as many positive integers as n
178)If x is a prime number greater than 5, y is a positive integer, and 5y=x^2+x, then y must be divisible by which of the following?
I. 5
II. 2x
III. x+1
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
179)If P is a prime number greater than 5, what is the remainder when P^2 is divided by 8.
A) 4
B) 3
C) 2
D) 1
E) Cannot be determined
180)Data Sufficiency->Is positive integer N divisible by 3?
(1) N^2/36 is an integer
(2) 144/N^2 is an integer
181)Data Sufficiency->If x is an integer, is x^3 divisible by 7?
(1) 3*x^12 is divisible by 7
(2) 3*x^4 is divisible by 7
182)Data Sufficiency->How many different prime factors does positive integer n have?
(1) 48< n^2 < 99
(2) 125n^2 has twelve factors
183)Data Sufficiency->Is the integer k divisible by 6?
(1) 6k is divisible by 12.
(2) 9k is divisible by 12.
184)If n = 3^8 - 2^8, which of the following is NOT a factor of n?
(A) 97
(B) 65
(C) 35
(D) 13
(E) 5
185)Data Sufficiency->If n is a positive integer, is n^3 – n divisible by 24 ?
(1) n = 2k + 1, where k is an integer.
(2) n^2 + n is divisible by 6
186)For which of the following values of n is (100+n)/n NOT an integer?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
187)Data Sufficiency->Is a*b*c divisible by 24?
(1) a, b, and c are consecutive even integers
(2) a*b is divisible by 12
188)Data Sufficiency->If x is the product of integers a, b, c, and d, is x divisible by 128?
(1) a = 24
(2) a, b, c, and d are consecutive even integers
189)Data Sufficiency->How many positive factors does positive integer N have?
1) N^3 has Four positive factors
2) 5N has four positive factors
190)Which of the following statements must be true?
A)If n is even then n^3-n is divisible by 24.
B)The sum of first 1000 prime numbers is even.
C)The product of first 32 primes is odd.
D)2 and 3 are the only consecutive prime numbers.
E)If n is odd then n^3-n is always divisible by 24.
191)Which of the following is NOT a factor of 10! ?
A. 1440
B. 625
C. 160
D. 80
E. 50
192)How many prime factors does the positive integer N have?
A)N^3 has 5 prime factors.
B)10N^3 has 7 Prime factors.
193)Data Sufficiency->How many prime factors does N have?
(1) N is a factor of 7200.
(2) 180 is a factor of N.
194)Data Sufficiency->If F is the prime factorization of N!, how many factors in F have an exponent of 1?
(1) 30 ≤ N ≤ 40
(2) 25 ≤ N ≤ 35
195)Data Sufficiency->Given that \(x\) is an integer and \(x\) is positive, is \(\frac{210}{x}\) also an integer??
(1) \(x\) is a prime number.
(2) \(x\) < 8
196)Which of the following is NOT a factor of the product of the first 50 positive multiples of 4 ?
A. \(17^2\)
B. \(11^4\)
C. \(7^6\)
D. \(47^{12}\)
E. \(2^{124}\)
197)Data Sufficiency->Is the integer \(n\) odd?
(1) a is an integer and \(n=a^7+a^5+a^3+a^2+2a+1\)
(2) 2n is divisible by twice as many positive integers as n
198)If a positive integer n has 211 factors,then how many prime factors does n have?
A)one
B)two
C)three
D)Four
E)cannot be determined
199)If a positive integer N has p factors ; how many factors will 2N have ?
A) p
B) 2p
C) P+1
D) 2p+1
E) Cannot be determined
200)Data Sufficiency ->Is integer k a prime number?
(1) k = 10! + m, where 1 < m < 8
(2) k is a multiple of 7
A)3
B)6
C)15
D)21
E)cannot be determined.
C.
102)If n is a positive integer and the product of all integers from 1 to n, inclusive, is a multiple of 990, what is the least possible value of n?
A)10
B)11
C)12
D)13
E)14
B.
Source-> Official Guide
Source-> Official Guide
103)n is a positive integer, and k is the product of all integers from 1 to n inclusive. If k is a multiple of 1440, then the smallest possible value of n is
A)8
B)12
C)6
D)18
E)24
A.
104)If x^2 is divisible by 216, what is the smallest possible value for positive integer x?
36.
105)If k is a positive even integer, what is the smallest possible value of k such that 3675*k is the square of an integer?
A.3
B.9
C.12
D.15
E.20
C.
NOTE->If you choose A=> Have a relook at the Question."k is an even integer" is specified in the Question-stem.
NOTE->If you choose A=> Have a relook at the Question."k is an even integer" is specified in the Question-stem.
106)If n = 4p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n ?
(A) Two
(B) Three
(C) Four
(D) Six
(E) Eight
C.
Easiest way is to put p=3 ->n=12 ->2,4,6,12 => Four even factors.
Alternatively -> \(n=2^2*p\) where p is an odd prime => for even factors => power of 2 must be at-least 1.
Possible cases => 2*2=> 4
Hence C.
Easiest way is to put p=3 ->n=12 ->2,4,6,12 => Four even factors.
Alternatively -> \(n=2^2*p\) where p is an odd prime => for even factors => power of 2 must be at-least 1.
Possible cases => 2*2=> 4
Hence C.
107)If N = \(2^2 * 3^3 * 5^5\),How many factors of N are divisible by 5 but not divisible by 3.
A)10
B)12
C)14
D)15
E)20
D.
For factors to be divisible by 5 but not divisible by 3=> Power of 3 must be zero and power of 5 must be greater than or equal to 1.
Possible cases => 3*1*5->15
Hence D.
For factors to be divisible by 5 but not divisible by 3=> Power of 3 must be zero and power of 5 must be greater than or equal to 1.
Possible cases => 3*1*5->15
Hence D.
108)What is the greatest prime factor of 12!11! + 11!10!?
(A) 7
(B) 11
(C) 13
(D) 17
(E) 19
E.
109)Data Sufficiency->How many distinct factors does positive integer k have?
(1) k has more distinct factors than the integer 9 but fewer distinct factors than the integer 81.
(2) k is the product of two distinct prime numbers.
D.
Statement 1-> 9 has 3 factors and 81 has 5 factors => k must have 4 factors => Sufficient.
Statement 2-> k=p*q for prime p and q => Number of factors => 2*2=>4=> Sufficient.
Hence D.
Statement 1-> 9 has 3 factors and 81 has 5 factors => k must have 4 factors => Sufficient.
Statement 2-> k=p*q for prime p and q => Number of factors => 2*2=>4=> Sufficient.
Hence D.
110) What is the greatest prime factor of (2^29) - (2^26)?
A)2
B)4
C)7
D)8
E)3
C.
(1) 44 < n^2 < 99
(2) 8n^2 has twelve factors
D.
Statement 1->
Notice the boundary condition for n is given.
Since n is an integer => n^2 must be a perfect square.
n^2=> 49,64 or 81
So n can be 7 or 2^3 or 3^2
In each case -> n will have just one prime factor.
Hence Sufficient.
Statement 2->
8n^12 has twelve factors.
2^3*n^2 has 12 factors => This is only possible if n is a prime number.
Thus it will have only one prime factor.
Hence D.
Source-> GMATClub-Tests
Statement 1->
Notice the boundary condition for n is given.
Since n is an integer => n^2 must be a perfect square.
n^2=> 49,64 or 81
So n can be 7 or 2^3 or 3^2
In each case -> n will have just one prime factor.
Hence Sufficient.
Statement 2->
8n^12 has twelve factors.
2^3*n^2 has 12 factors => This is only possible if n is a prime number.
Thus it will have only one prime factor.
Hence D.
Source-> GMATClub-Tests
112)What is the greatest prime factor of 9919?
A) 7
B) 13
C) 17
D) 97
E) 109
E.
9919=100^2-9^2=> 109*91=>109*7*13.
Notice that 109 is a prime number.
Hence the greatest prime factor ->109
Hence E.
9919=100^2-9^2=> 109*91=>109*7*13.
Notice that 109 is a prime number.
Hence the greatest prime factor ->109
Hence E.
113)If 5x^2 has two different prime factors, at most how many different prime factors does x have?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
B.
Two properties are been tested here =>
One-> \(X\) and \(X^n\) always have the exact same prime factors.
Two->When we multiply any positive integer x with another positive integer n -> The number of prime factors of nx may be greater than or equal to x.It depends on the value of n.
So \(x\) and \(x^2\) will have the exact same primes.
Two cases are possible here =>
Case 1=>
\(x\) has one prime factor which is not equal to 5.
So 5\(x^2\) will have 2 primes.
Case 2->
\(x\) has two prime factors,one out of which is 5 so that 5x^2 has two prime factors.
Hence almost \(x\) can have 2 prime factors.
Hence B.
Source->GMAT-prep
Two properties are been tested here =>
One-> \(X\) and \(X^n\) always have the exact same prime factors.
Two->When we multiply any positive integer x with another positive integer n -> The number of prime factors of nx may be greater than or equal to x.It depends on the value of n.
So \(x\) and \(x^2\) will have the exact same primes.
Two cases are possible here =>
Case 1=>
\(x\) has one prime factor which is not equal to 5.
So 5\(x^2\) will have 2 primes.
Case 2->
\(x\) has two prime factors,one out of which is 5 so that 5x^2 has two prime factors.
Hence almost \(x\) can have 2 prime factors.
Hence B.
Source->GMAT-prep
114)What is the greatest prime factor of 12!11! + 11!10!?
(A) 7
(B) 11
(C) 13
(D) 17
(E) 19
E.
12!11! + 11!10!=> 11!10![12*11+1]=>11!10!*133=>11!10!*7*19
Clearly the greatest prime factor will be 19.
Hence E.
12!11! + 11!10!=> 11!10![12*11+1]=>11!10!*133=>11!10!*7*19
Clearly the greatest prime factor will be 19.
Hence E.
115)Data Sufficiency->How many different prime numbers are factors of positive integer n?
(1) 4 different prime numbers are factors of \(2n\).
(2) 4 different prime numbers are factors of \(n^2\).
B.
Properties in action->
\(X\) and \(X^n\) always have the exact same prime factors.
When we multiply any positive integer x with another positive integer n -> The number of prime factors of nx may be greater than or equal to x.It depends on the value of n.
Statement 1=>
From this statement we can conclude that n can have 4 prime factors or 3 prime factors.
Hence not sufficient.
Statement 2=>
From this statement it is evident that n must have exactly 4 prime factors.
Hence Sufficient.
Hence B.
Source-> GMAT-Prep
Properties in action->
\(X\) and \(X^n\) always have the exact same prime factors.
When we multiply any positive integer x with another positive integer n -> The number of prime factors of nx may be greater than or equal to x.It depends on the value of n.
Statement 1=>
From this statement we can conclude that n can have 4 prime factors or 3 prime factors.
Hence not sufficient.
Statement 2=>
From this statement it is evident that n must have exactly 4 prime factors.
Hence Sufficient.
Hence B.
Source-> GMAT-Prep
116)What is the greatest prime factor of 3^6 - 1 ?
A. 2
B. 3
C. 7
D. 13
E. 17
3^6-1=> 27^2-1=> (27+1)*(27-1)=> 2^3*7*13
Clearly 13 is the greatest prime factor.
Hence D.
Clearly 13 is the greatest prime factor.
Hence D.
117)Data Sufficiency->If m and n are different positive integers, then how many prime numbers are in set {m, n, m + n}?
(1) mn is prime.
(2) m + n is even.
C.
In this Question,we are told that m and n are different positive integers and we are asked as to how many numbers out of {m, n, m + n} are prime.
Statement 1=>
m*n is a prime number
This can happen only when one out of m or n must be 1 and the other must be a prime number.
Case 1 => When an even prime i.e 2 is involved -> {1,2,3} => two numbers in this set will be prime.
Case 2=> When an odd prime is involved -> {1,odd-prime,even number >2}=> Only one prime number.
Hence not sufficient.
Statement 2->m+n is even
e.g=> {3,5,8} => two prime
or {2,4,6}=> None are prime => Not sufficient.
Combining the two statements => m+n is even and one out of the is one => The other must be odd prime.
Thus in the set => {1,Odd-prime,Even number>1}=> Only one number will be prime.
Hence C.
In this Question,we are told that m and n are different positive integers and we are asked as to how many numbers out of {m, n, m + n} are prime.
Statement 1=>
m*n is a prime number
This can happen only when one out of m or n must be 1 and the other must be a prime number.
Case 1 => When an even prime i.e 2 is involved -> {1,2,3} => two numbers in this set will be prime.
Case 2=> When an odd prime is involved -> {1,odd-prime,even number >2}=> Only one prime number.
Hence not sufficient.
Statement 2->m+n is even
e.g=> {3,5,8} => two prime
or {2,4,6}=> None are prime => Not sufficient.
Combining the two statements => m+n is even and one out of the is one => The other must be odd prime.
Thus in the set => {1,Odd-prime,Even number>1}=> Only one number will be prime.
Hence C.
118)What is the greatest prime factor of 1+2+3+。。。+40?
A. 17
B. 29
C. 31
D. 37
E. 41
E.
Sum => 1+2+3....40=> 20*41 => 2^2*5*41
41 is clearly the greatest prime factor.
Hence E.
Sum => 1+2+3....40=> 20*41 => 2^2*5*41
41 is clearly the greatest prime factor.
Hence E.
119)Data Sufficiency->How many positive factors does positive integer \(N\) have?
1) \(N^2\) has three positive factors.
2) \(2N\) has four positive factors.
A.
We are asked about the number of factors of a positive integer N.
Statement 1->
\(N^2\) has 3 factors
The only way \(N^2\) can have 3 factors is if N is a prime number.
So \(N\) must have 2 factors.
Hence Sufficient.
Statement 2->
\(2N\) has 4 Factors
Two cases are possible.
Case 1=>
\(N\) is a prime number.=> \(N\) will have 2 factors
Case 1->
\(N=2^2\)=> \(N\) will have 3 factors.
Hence Not sufficient.
Hence A.
We are asked about the number of factors of a positive integer N.
Statement 1->
\(N^2\) has 3 factors
The only way \(N^2\) can have 3 factors is if N is a prime number.
So \(N\) must have 2 factors.
Hence Sufficient.
Statement 2->
\(2N\) has 4 Factors
Two cases are possible.
Case 1=>
\(N\) is a prime number.=> \(N\) will have 2 factors
Case 1->
\(N=2^2\)=> \(N\) will have 3 factors.
Hence Not sufficient.
Hence A.
120)If \(n=3*4*p\) where p is a prime number greater than 3,how many different positive non-prime divisors does n have, excluding 1 and n?
(A) Six
(B) Seven
(C) Eight
(D) Nine
(E) Ten
B.
\(n=3*2^2*p\)
n will therefore have 12 factors including1,n and its prime factors 2,3,p
So excluding them all n will have -> 12-5=7 factors.
Hence B
\(n=3*2^2*p\)
n will therefore have 12 factors including1,n and its prime factors 2,3,p
So excluding them all n will have -> 12-5=7 factors.
Hence B
121)Data Sufficiency->How many divisors does the positive integer \(N\) have?
(1)\(27N^3\) has 16 factors.
(2)\(90<N^3<200\)
D.
In statement 1 we can collude that N is a prime number.
In statement 2 => N must be 5.
Hence D.
In statement 1 we can collude that N is a prime number.
In statement 2 => N must be 5.
Hence D.
122)Data Sufficiency-> How many prime factors does positive integer n have?
(1) n/5 has only a prime factor.
(2) 3*n^2 has two different prime factors.
E.
Taking test cases =>
n=5^3
n=5*3
We can choose E.
Taking test cases =>
n=5^3
n=5*3
We can choose E.
123)What is the greatest prime factor of \(2^{10}*5^4 - 2^{13}*5^2 + 2^{14}\)?
(A) 2
(B) 3
(C) 7
(D) 11
(E) 13
C.
124)Data Sufficiency->If p is a positive integer, is 2p + 1 prime?
(1) p is prime.
(2) units digit of p is not prime.
E.
125)What is the greatest prime factor of \(6^8−3^8\) ?
A) 3
B) 11
C) 17
D) 19
E) 31
C.
Proper in action=> \(A^2-B^2=(A+B)*(A-B)\)
Proper in action=> \(A^2-B^2=(A+B)*(A-B)\)
126)Data Sufficiency->If \(p\) is a positive integer, is \(p\) a prime number?
(1) \(p\) and \(p+1\) have the same number of factors.
(2) \(p-1\) is a factor of \(p\).
B.
We need to check if p is prime or not.
Statement 1->B.
Lets use some test cases =>
p=2
p+1=3
Both have 2 factors as both are prime.
Hence p is prime
But wait.Lets find some other test case for which p is non prime.
p=21
p+1=22
Both have 4 factors
And p is clearly non prime.
Another case be 14 and 15
Hence not sufficient.
Actually there exist ∞ such cases.
Statement 2->
For every positive integer >1 p and p-1 are always co-prime.
The only value of p possible is 2.
Hence p=2,which is a prime number.
Hene B
We need to check if p is prime or not.
Statement 1->B.
Lets use some test cases =>
p=2
p+1=3
Both have 2 factors as both are prime.
Hence p is prime
But wait.Lets find some other test case for which p is non prime.
p=21
p+1=22
Both have 4 factors
And p is clearly non prime.
Another case be 14 and 15
Hence not sufficient.
Actually there exist ∞ such cases.
Statement 2->
For every positive integer >1 p and p-1 are always co-prime.
The only value of p possible is 2.
Hence p=2,which is a prime number.
Hene B
127)Data Sufficiency->If p is a positive integer, is integer k prime?
(1) 3p + 3 = k
(2) 7! + 3 = k
D.
128)Data Sufficiency->k is a positive integer. Is k prime?
(1) At least one number in the set {1, k, k + 7} is prime.
(2) k is odd.
C.
From statement 1->Using test cases=> k =4 or k=3
Hence not sufficient.
From statement 2=> k=3 or k=15
Hence not sufficient
Combing them => k+7 will be an even number >2, so it cannot be prime.
Hence k must be prime so that one out of 1,k,k+7 is prime.
Hence C.
From statement 1->Using test cases=> k =4 or k=3
Hence not sufficient.
From statement 2=> k=3 or k=15
Hence not sufficient
Combing them => k+7 will be an even number >2, so it cannot be prime.
Hence k must be prime so that one out of 1,k,k+7 is prime.
Hence C.
129)Data Sufficiency->What is the value of the integer p?
(1) p is a prime number.
(2) 88 ≤ p ≤ 95
C.
Note->Only prime value in the boundary mentioned is 89.
Note->Only prime value in the boundary mentioned is 89.
130)Data Sufficiency->If n is a positive integer, does n have four or more distinct factors?
(1) n is not prime
(2)900 ≤ n < 1100
E.
Note-> For combination statement=>
Two examples for n=31^2=> n has less than 4 factors
or n=900=> n has more than 4 factors
Note-> For combination statement=>
Two examples for n=31^2=> n has less than 4 factors
or n=900=> n has more than 4 factors
131)If x is an integer and x^2 is even, which of the following must be true?
I. x is odd.
II. x is even.
III. x^3 is odd.
(A) I only
(B) II only
(C) III only
(D) I and II only
(E) II and III only
B.
Property in action=> x and x^n always have the same even/odd nature if n≠0
Property in action=> x and x^n always have the same even/odd nature if n≠0
132)Data Sufficiency->If K is a positive 3-digit number, is K prime?
(1) The last digit of K is not even
(2) K is the smallest number possible where the hundreds digit is the sum of the tens and units digit. Tens and units digits are equal.
B.
Statement 1 is off course nor sufficient.
We can use test cases => 211 is prime and 145 is not prime.
Notice that from statement 2 => K must be 211.
And 211 is a prime number as it is not divisible by any primes less than or equal to the square root of 211.
Hence B.
Statement 1 is off course nor sufficient.
We can use test cases => 211 is prime and 145 is not prime.
Notice that from statement 2 => K must be 211.
And 211 is a prime number as it is not divisible by any primes less than or equal to the square root of 211.
Hence B.
133)Data Sufficiency->If x and y are positive integers, is y divisible by 3?
(1) y = 2x^3 + 9x^2 - 5x.
(2) x is an odd number
A.
It is all about patterns.
Here is what i did in this Question=>
We need to see if y/3 is an integer or not.
We are given that a and y are positive integers.
Statement 1=>
\(y = 2x^3 + 9x^2 - 5x\)
Taking out x as a factor we get -> \(x[2x^2+9x-5]=> x(2x-1)(x+5)\)
Now putting in the values of x=> y is always a multiple of 3.
Hence sufficient.
Statement 2=>
No clue of y=> Not sufficient.
Hence A.
It is all about patterns.
Here is what i did in this Question=>
We need to see if y/3 is an integer or not.
We are given that a and y are positive integers.
Statement 1=>
\(y = 2x^3 + 9x^2 - 5x\)
Taking out x as a factor we get -> \(x[2x^2+9x-5]=> x(2x-1)(x+5)\)
Now putting in the values of x=> y is always a multiple of 3.
Hence sufficient.
Statement 2=>
No clue of y=> Not sufficient.
Hence A.
134)If \(x\) and \(y\) are positive integer and \(xy\) is divisible by 4, which of the following must be true?
A) If \(x\) is even then \(y\) is odd.
B) If \(x\) is odd then \(y\) is a multiple of 4.
C) If \(x+y\) is odd then \(y/x\) is not an integer.
D) If \(x+y\) is even then \(x/y\) is an integer.
E) \(x^y\) is even.
B.
135)If n is a prime number greater than 3, what is the remainder when n^2 is divided by 12 ?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 5
B.
Source->Official Guide
Source->Official Guide
136)Data Sufficiency->If x is a prime number, what is the value of x?
(1) 2x + 2 is the cube of a positive integer.
(2) The average of any x consecutive integers is an integer.
E.
We are told that x is a prime number and asked about its value.
Statement 1->
2x+2=t^3
for any positive integer t.
Lets use some test cases =>
2x+2=1
x=1/2=> Not allowed
2x+2=8
x=3=> Allowed
2x+2=27
x=25/2=> Not allowed
2x+2=64
x=31=>Allowed
Hence 3 and 31 are both acceptable values => Not sufficient.
Statement 2->
Remember for set of consecutive integers is an AP series.
Mean can be of the form p if x is odd (p is an integer )
Mean can be of the form p.5 if x is even (p is an integer)
RULE->Sum of n consecutive integers is always divisible by n for n being odd and never divisible by n for n being even.
Hence this statement tells us that x is odd.
But their are ∞ odd primes => Not sufficient.
Combing them x=3 and x=31 are both acceptable values.
Hence not sufficient .
Hence E.
We are told that x is a prime number and asked about its value.
Statement 1->
2x+2=t^3
for any positive integer t.
Lets use some test cases =>
2x+2=1
x=1/2=> Not allowed
2x+2=8
x=3=> Allowed
2x+2=27
x=25/2=> Not allowed
2x+2=64
x=31=>Allowed
Hence 3 and 31 are both acceptable values => Not sufficient.
Statement 2->
Remember for set of consecutive integers is an AP series.
Mean can be of the form p if x is odd (p is an integer )
Mean can be of the form p.5 if x is even (p is an integer)
RULE->Sum of n consecutive integers is always divisible by n for n being odd and never divisible by n for n being even.
Hence this statement tells us that x is odd.
But their are ∞ odd primes => Not sufficient.
Combing them x=3 and x=31 are both acceptable values.
Hence not sufficient .
Hence E.
137)The sum of prime numbers that are greater than 60 but less than 70 is
(A) 67
(B) 128
(C) 191
(D) 197
(E) 260
B.
Only 2 prime are present in the boundary -> (60,70) => 61 and 67.
Sum =128.
Hence B.
Source->Official Guide
Only 2 prime are present in the boundary -> (60,70) => 61 and 67.
Sum =128.
Hence B.
Source->Official Guide
138)Data Sufficiency->If x, y, and z are positive integers, what is the greatest prime factor of the product xyz?
(1) The greatest common factor of x, y, and z is 7.
(2) The lowest common multiple of x, y, and z is 84.
B.
Given data -> x,y,z are positive integers.
We are asked about the Greatest prime factor of x*y*z
Statement 1->
GCD(x,y,z)=7
Lets use test cases->
Case 1=>
7
7
7
GCD=> 7
And Greatest prime =7
Case 2=>
7
7
7*13
GCD=>7
Greatest prime factor is 7*13
Actually this statement tells us that the greatest prime factor must be greater than or equal to 7.
Hence not sufficient.
Statement 2->
LCM(x,y,z)=84=2^2*3*7
Now if x,y,z had any prime factor other than 2,3,7 => It must have been present in the LCM.
In LCM => We pick the greatest power of each Prime factor.
Hence the greatest prime in the Product must be 7.
Hence sufficient.
Hence B.
Given data -> x,y,z are positive integers.
We are asked about the Greatest prime factor of x*y*z
Statement 1->
GCD(x,y,z)=7
Lets use test cases->
Case 1=>
7
7
7
GCD=> 7
And Greatest prime =7
Case 2=>
7
7
7*13
GCD=>7
Greatest prime factor is 7*13
Actually this statement tells us that the greatest prime factor must be greater than or equal to 7.
Hence not sufficient.
Statement 2->
LCM(x,y,z)=84=2^2*3*7
Now if x,y,z had any prime factor other than 2,3,7 => It must have been present in the LCM.
In LCM => We pick the greatest power of each Prime factor.
Hence the greatest prime in the Product must be 7.
Hence sufficient.
Hence B.
139)If n=6p, where p is a prime number greater than 2, how many different positive even divisors does n have, including n?
A)2
B)3
C)4
D)6
E)cannot be determined.
E.
Two cases exist here=>
Case 1-> p=3=> Even divisors ->3
Case 2-> p=prime >3 => even divisors ->4
Hence E.
Two cases exist here=>
Case 1-> p=3=> Even divisors ->3
Case 2-> p=prime >3 => even divisors ->4
Hence E.
140)Given that N=a^3*b^4*c^5 where a, b and c are distinct prime numbers, what is the smallest number with which N should be multiplied such that it becomes a perfect square, a perfect cube as well as a perfect fifth power?
A. a^3*b^4*c^5
B. a^5*b^4*c^3
C. a^2*b^3*c^5
D. a^7*b^6*c^5
E. a^27*b^26*c^25
E.
Perfect square => Powers of all primes=> Multiple of 2.
Perfect cube=>Powers of all the primes => Multiple of 3.
Perfect fifth power => Powers of all the primes => Multiple of 5
Hence powers of all the primes must be a multiple of all->2,3,5 => LCM=30
Smallest multiplication => \(a^{27}*b^{26}*c^{25}\)
Hence E.
Perfect square => Powers of all primes=> Multiple of 2.
Perfect cube=>Powers of all the primes => Multiple of 3.
Perfect fifth power => Powers of all the primes => Multiple of 5
Hence powers of all the primes must be a multiple of all->2,3,5 => LCM=30
Smallest multiplication => \(a^{27}*b^{26}*c^{25}\)
Hence E.
141)Data Sufficiency->Is the integer b divisible by 6 ?
(1) 8b is divisible by 3.
(2) 9b is divisible by 12.
C.
We need to see if B is divisible by 6 or not.
Statement 1=> From this statement we can conclude that B must be divisible by 3.
E.g-> 3,6
Not sufficient.
Statement 2=> Form this statement , we can conclude that B must be divisible by 4
E.g->4,4*6
Not sufficient.
Combining the two statement => B must be divisible by 12.
Hence it must be divisible by 6 too.
Hence C.
We need to see if B is divisible by 6 or not.
Statement 1=> From this statement we can conclude that B must be divisible by 3.
E.g-> 3,6
Not sufficient.
Statement 2=> Form this statement , we can conclude that B must be divisible by 4
E.g->4,4*6
Not sufficient.
Combining the two statement => B must be divisible by 12.
Hence it must be divisible by 6 too.
Hence C.
142)Data Sufficiency->If k is a positive integer, is k a prime number?
(1) No integers between 2 and \(\sqrt{k}\), inclusive divides k evenly.
(2) No integers between 2 and k/2 inclusive divides k evenly, and k is greater than 5.
D.
In order to check whether a given number is prime or not =>We must check its divisibility with all the prime numbers less than or equal to the square root of that number.
So clearly A is sufficient.
Statement 2 is a bit a trickier.
Lets compare the values of √k and k/2 for integers k>5
10=> √10=3.something 10/2=5
100=> √100=10 100/2=50
10000=> √10000=100 10000/2=5000
Clearly as the number is increasing the gap between √k and k/2 is also increasing.
Hence k must not have any prime factors between 1 and √k too.[/highlight]
Thus k is a prime number.
Hence Sufficient.
Hence D.
In order to check whether a given number is prime or not =>We must check its divisibility with all the prime numbers less than or equal to the square root of that number.
So clearly A is sufficient.
Statement 2 is a bit a trickier.
Lets compare the values of √k and k/2 for integers k>5
10=> √10=3.something 10/2=5
100=> √100=10 100/2=50
10000=> √10000=100 10000/2=5000
Clearly as the number is increasing the gap between √k and k/2 is also increasing.
Hence k must not have any prime factors between 1 and √k too.[/highlight]
Thus k is a prime number.
Hence Sufficient.
Hence D.
143)The "prime sum" of an integer n greater than 1 is the sum of all the prime factors of n, including repetitions. For example , the prime sum of 12 is 7, since 12 = 2 x 2 x 3 and 2 +2 + 3 = 7. For which of the following integers is the prime sum greater than 35 ?
(A) 440
(b) 512
(C) 620
(D) 700
(E) 750
C.
Source->Official-Guide
Source->Official-Guide
144)Data Sufficiency->If a and b are integers, is a + b divisible by 15?
(1) a + b is divisible by 30.
(2) a + b is divisible by 5
A.
145)Data Sufficiency->If A and B are positive integers, is B divisible by A?
(1) 2B/A is an integer.
(2) B^2/A is an integer
E.
We are told that A and B are positive integers and we are asked if if B/A is an integer or not.
Lets use some test cases here.
Statement 1->
2B/A is an integer.
Let A=1 and B=1 => 2B/A is an integer and B/A =1 which is an integer.
Let A=2 and B=1 => 2B/A is an integer and B/A=1/2 which is not an integer.
Hence not sufficient.
Statement 2->
B^2/A is an integer.
Let A=1 and B=1 => B^2/A is an integer and B/A=1 which is an integer.
Let A=4 and B=2 => B^2/A is an integer and B/A=1/2 which is not an integer.
Hence not sufficient.
Combing the two statements ->
Let A=1 and B=1 => B/A is an integer.
Let A=4 and B=2 => B/A is not an integer.
Hence not sufficient.
Hence E.
We are told that A and B are positive integers and we are asked if if B/A is an integer or not.
Lets use some test cases here.
Statement 1->
2B/A is an integer.
Let A=1 and B=1 => 2B/A is an integer and B/A =1 which is an integer.
Let A=2 and B=1 => 2B/A is an integer and B/A=1/2 which is not an integer.
Hence not sufficient.
Statement 2->
B^2/A is an integer.
Let A=1 and B=1 => B^2/A is an integer and B/A=1 which is an integer.
Let A=4 and B=2 => B^2/A is an integer and B/A=1/2 which is not an integer.
Hence not sufficient.
Combing the two statements ->
Let A=1 and B=1 => B/A is an integer.
Let A=4 and B=2 => B/A is not an integer.
Hence not sufficient.
Hence E.
146)Data Sufficiency->If n is a positive integer is n-1 divisible by 3 ?
(1) n^2+n is not divisible by 6.
(2) 3n=3k+3 where k is a positive multiple of 3
D.
147)Data Sufficiency->If n is a positive integer, is n – 1 divisible by 3?
(1) n^2 + n is not divisible by 6.
(2) 3n=k+3 where k is a positive multiple of 3.
A.
148)Suppose x is the product of all the primes less than or equal to 59. How many primes appear in the set {x + 2, x + 3, x + 4, …, x + 59}?
A)0
B)17
C)18
D)23
E)24
A.
149)The smallest prime factor of 899 is \(x\). Which of the following is true of \(x\)?
A. \(1 \lt x \le 7\)
B. \(7 \lt x \le 14\)
C. \(14 \lt x \le 21\)
D. \(21 \lt x \le 28\)
E. \(28 \lt x \le 35\)
E.
Notice 899=900-1=> 30^2-1=> 29*31
Both 29 and 31 are primes.
Hence smallest prime factor -> 29.
Hence E.
Notice 899=900-1=> 30^2-1=> 29*31
Both 29 and 31 are primes.
Hence smallest prime factor -> 29.
Hence E.
150)Data Sufficiency->If n is a positive integer, does n have four or more distinct factors?
(1) n is not prime
(2) 150 ≤ n < 200
E.
We need to see if the number of factors of n is greater than or equal to 4 or not.
A few key links =>
1 is the only number that has one factor.
A prime number has exactly two factors.
For any number to have 3 factors it must be of the form => Prime^2 i.e square of a prime number.
Lets get on with the statements=>
Statement 1->
n is not prime.
Lets play around with some test cases.
n=1 => One factor => n has less than 4 factors
n=200=> 2^3*5^2 => 12 factors => n has more than 4 factors.
Hence insufficient.
Statement 2->
n lies in the range [150,200]
Again lets use some test cases here.
n=151=> Prime number as it is not divisible by any prime number less than equal to the square root of 151.
So n has 2 factors which is less than 4.
n=200=> 12 factors => more than 4 factors.
Not sufficient.
Combing the two statements =>
n is not prime.
ans n≠1
So n can never has one or two factors.
But what about three?
Can n be square of a prime ?
YES for n=13^2=169=> n has 3 factors which is less than 4
For all values of n => n will have more than 3 factors.
Hence not sufficient.
Hence E.
We need to see if the number of factors of n is greater than or equal to 4 or not.
A few key links =>
1 is the only number that has one factor.
A prime number has exactly two factors.
For any number to have 3 factors it must be of the form => Prime^2 i.e square of a prime number.
Lets get on with the statements=>
Statement 1->
n is not prime.
Lets play around with some test cases.
n=1 => One factor => n has less than 4 factors
n=200=> 2^3*5^2 => 12 factors => n has more than 4 factors.
Hence insufficient.
Statement 2->
n lies in the range [150,200]
Again lets use some test cases here.
n=151=> Prime number as it is not divisible by any prime number less than equal to the square root of 151.
So n has 2 factors which is less than 4.
n=200=> 12 factors => more than 4 factors.
Not sufficient.
Combing the two statements =>
n is not prime.
ans n≠1
So n can never has one or two factors.
But what about three?
Can n be square of a prime ?
YES for n=13^2=169=> n has 3 factors which is less than 4
For all values of n => n will have more than 3 factors.
Hence not sufficient.
Hence E.
151)Data Sufficiency->If p is a positive integer, what is the value of p?
(1) p/4 is a prime number.
(2) p is divisible by 3
C.
Combing the two statements => p must be 12.
Source->Gmat-Prep
Combing the two statements => p must be 12.
Source->Gmat-Prep
152)Data Sufficiency->If a and b are integers, is (ab+2)(ab+3)(ab+4) divisible by 12?
(1) a=even
(2) b=odd
A.
153)An integer n that is greater than 1 is said to be "prime-saturated" if it has no prime factor greater than or equal to √n.
Which of the following integers is prime saturated?
A) 6
B) 35
C) 46
D) 66
E) 75
E.
Source->Gmat-Prep
Source->Gmat-Prep
154)Data Sufficiency->If p is a positive integer, is p^2 divisible by 96?
(1) p is a multiple of 8.
(2) p^2 is a multiple of 12
C.
155)If x is a positive integer greater than 1, what is the sum of the multiples of x from x to x^2, inclusive?
(A) x(x + 1)(x-1)
(B) x^2(x + 1)/2
(C) x^2(x-1)
(D) (x^3 + 2x)/2
(E) x(x-1)^2
B.
Plugging in x=5 => Sum =75
Only option that matches is B.
Hence B.
Note=> While plugging in numbers we must check for each option.There are cases when you might get more than one answer.
In such cases we must alter our test cases to choose between the remaining options.
Plugging in x=5 => Sum =75
Only option that matches is B.
Hence B.
Note=> While plugging in numbers we must check for each option.There are cases when you might get more than one answer.
In such cases we must alter our test cases to choose between the remaining options.
156)Data Sufficiency->Does p^2 = q if p is a prime number?
(1) q^2 – p^2 =0
(2) p^2 = 49
A.
157)A number is said to be prime saturated if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest two digit prime saturated integer ?
A. 99
B. 98
C. 97
D. 96
E. 95
D.
Source-> GMAT-prep
Source-> GMAT-prep
158)Data Sufficiency->How many different prime factors does x have?
1) \(5x^2\) has two different prime factors
2) \(x > 1\)
E.
Key Properties applicable in this question =>
X and X^n always have the exact same prime factors.
When we multiple a positive integer x with some other positive integer n => The number of prime factors of nx may be greater than or equal to the number of prime factors of x.
We need the number of prime factors of x.
Statement 1->
5x^2 has primes
x can have 2 primes (if 5 is one of its primes)
x can have 1 prime (if 5 is not one of its primes)
Hence not sufficient.
Statement 2->
x>1
Clearly not sufficient.
Combing the two statements =>
Still not sufficient.
E.g =>
x=3
or
x=5*3
Satisfy both the equations.
Hence E.
Key Properties applicable in this question =>
X and X^n always have the exact same prime factors.
When we multiple a positive integer x with some other positive integer n => The number of prime factors of nx may be greater than or equal to the number of prime factors of x.
We need the number of prime factors of x.
Statement 1->
5x^2 has primes
x can have 2 primes (if 5 is one of its primes)
x can have 1 prime (if 5 is not one of its primes)
Hence not sufficient.
Statement 2->
x>1
Clearly not sufficient.
Combing the two statements =>
Still not sufficient.
E.g =>
x=3
or
x=5*3
Satisfy both the equations.
Hence E.
159)How many different prime factors does positive integer x have?
(1) \(1 < x < 6\)
(2) \(5x^2\) has four factors.
D.
We need the number of prime factors of the positive integer x.
Statement 1=>
x=>(1,6)=> x=2,3,4,5=> In each case => x will have just one prime factor=> Sufficient.
Statement 2=> The only value of x possible is 5.
Hence x will have only one factor. Sufficient
Hence D.
We need the number of prime factors of the positive integer x.
Statement 1=>
x=>(1,6)=> x=2,3,4,5=> In each case => x will have just one prime factor=> Sufficient.
Statement 2=> The only value of x possible is 5.
Hence x will have only one factor. Sufficient
Hence D.
160)Data Sufficiency->If N is a positive integer, does N have exactly three factors?
(1) The integer N^2 has exactly five factors
(2) Only one factor of N is a prime number
A.
We need to check if n has 3 factors or not.
Firstly a few key kicks =>
1 is the only number that has one factor.
A prime has exactly 2 factors.
For a number to have 3 factors =>It must be of the form => Prime^2 i.e square of a prime number.
Lets jump into Statements.
Statement 1->
n^2 has 5 factors.
This can only happen if n=Prime^2
Hence sufficient.
Statement 2->
This statement is basically saying that n has only prime factor.
But it does not tell us anything about the exponent of that prime.
E.g=> n=2^2=> 3 factors
n=2^100=>101 factors
Hence not sufficient.
Hence A.
We need to check if n has 3 factors or not.
Firstly a few key kicks =>
1 is the only number that has one factor.
A prime has exactly 2 factors.
For a number to have 3 factors =>It must be of the form => Prime^2 i.e square of a prime number.
Lets jump into Statements.
Statement 1->
n^2 has 5 factors.
This can only happen if n=Prime^2
Hence sufficient.
Statement 2->
This statement is basically saying that n has only prime factor.
But it does not tell us anything about the exponent of that prime.
E.g=> n=2^2=> 3 factors
n=2^100=>101 factors
Hence not sufficient.
Hence A.
161)Data Sufficiency->What is the value of x?
(1) x is the square of an integer.
(2) 577 < x < 675
C.
162)Data Sufficiency->If x is a prime number, what is the value of x.
1)There are 4 prime numbers between 11 and x
2 )There is no y such that 1<y<x and y is the divisor of x.
A.
163)Data Sufficiency->If x is a prime number, what is the value of x.
1)There are 3 prime numbers between 11 and x
2 )x+1 is a prime number
B.
164)For how many positive integers is the number of positive divisors equal to the number itself ?
A) none
B) one
C) two
D)three
E)cannot be determined
C.
One and two are only number for which the number of factors is equal to the number itself.
Hence C.
One and two are only number for which the number of factors is equal to the number itself.
Hence C.
165) The positive integer 2000 has how many factors?
A)2
B)10
C)12
D)16
E)20
E.
166)Data Sufficiency->If x is a perfect square greater than 1, what is the value of x?
(1) x has exactly 3 distinct factors.
(2) x has exactly one positive odd factor
C.
We are given that x is a perfect square and are asked its value.
Statement 1->
x has 3 factors.
Some quick facts on factors ->
1 is the only number that has 1 factor.
A prime as 2 factors.
A number of the form Prime^2 has 3 factors.
Hence x must be of the form Prime^2
E.g=> 2^2 or 3^2 etc.
Hence not sufficient.
Statement->2
As 1 is the factor of every number => This statement tells us that x must be even.
E.g => 2^2 or 2^4 or 2^6
Basically it cannot have any prime factor other then one.
So it must be of the form 2^even number .
Hence not sufficient.
Combing the two statements=>
x must be 2^2 to have 4 factors.
Hence sufficient.
Hence C.
We are given that x is a perfect square and are asked its value.
Statement 1->
x has 3 factors.
Some quick facts on factors ->
1 is the only number that has 1 factor.
A prime as 2 factors.
A number of the form Prime^2 has 3 factors.
Hence x must be of the form Prime^2
E.g=> 2^2 or 3^2 etc.
Hence not sufficient.
Statement->2
As 1 is the factor of every number => This statement tells us that x must be even.
E.g => 2^2 or 2^4 or 2^6
Basically it cannot have any prime factor other then one.
So it must be of the form 2^even number .
Hence not sufficient.
Combing the two statements=>
x must be 2^2 to have 4 factors.
Hence sufficient.
Hence C.
167)Data Sufficiency->How many factors does y have?
(1) y is the cube of an integer.
(2) y is the product of 2 distinct positive digits.
C.
We need to get the number of factors of y.
Statement 1->
y is a perfect cube.
y can be negative or zero or positive.
It would have been nice if the Question Stem had read-> y is a polite integer.
Anyways its clearly not possible to get the factors.
Not sufficient.
Statement 1->
From this statement we can infer that y must be positive.
y=2*3=> 6(4 factors)
y=1*5=>5(two factors)
y=3*9=>27=> 4 factors.
Hence not sufficient.
Combing the two statements=> The only values that y can take are 8 and 27 and they both have 4 factors.
Hence C.
We need to get the number of factors of y.
Statement 1->
y is a perfect cube.
y can be negative or zero or positive.
It would have been nice if the Question Stem had read-> y is a polite integer.
Anyways its clearly not possible to get the factors.
Not sufficient.
Statement 1->
From this statement we can infer that y must be positive.
y=2*3=> 6(4 factors)
y=1*5=>5(two factors)
y=3*9=>27=> 4 factors.
Hence not sufficient.
Combing the two statements=> The only values that y can take are 8 and 27 and they both have 4 factors.
Hence C.
168)Data Sufficiency->If x, y and k are integers, is xy divisible by 3?
(1) y = 2^(16) - 1
(2) The sum of the digits of x equals 6^k
A.
Note->k can be zero too.
Note->k can be zero too.
(1) x is less than 15
(2) (x-2) is multiple of 5
E.
Note-> x=2 and x=7 satisfy both the statements.
Note-> x=2 and x=7 satisfy both the statements.
170)Data Sufficiency->How many prime factors does positive integer n have?
(1) n/7 has only one prime factor.
(2) 3*n^2 has two different prime factors.
E.
Note-> Values such as x=7^2 and 7*3 will satisfy both the statements.
Hence E.
Note-> Values such as x=7^2 and 7*3 will satisfy both the statements.
Hence E.
171)If x = 13y, where y is a prime number greater than 2, how many different positive even divisors does x have, including x?
A. 0
B. 1
C. 2
D. 3
E. It cannot be determined from the information given
A.
172)Data Sufficiency ->If b is a positive integer, then b has how many distinct positive factors?
(1) all of b's factors are also factors of 62.
(2) b is the product of one even prime number and one odd prime number.
B.
We need to get the number of factors of b.
Statement 1->
This is just another way of saying that 62/b will be an integer i.e b is a factor of 62.
b=2
b=1
b=62
Etc .
Hence not sufficient.
Statement 2=>
b=Product of two prime numbers that are different.
Hence b will be of the from prime1 *prime2
Hence number of factors will be 4.
Hence sufficient.
Hence B.
We need to get the number of factors of b.
Statement 1->
This is just another way of saying that 62/b will be an integer i.e b is a factor of 62.
b=2
b=1
b=62
Etc .
Hence not sufficient.
Statement 2=>
b=Product of two prime numbers that are different.
Hence b will be of the from prime1 *prime2
Hence number of factors will be 4.
Hence sufficient.
Hence B.
173)Data Sufficiency->How many different factors does the integer n have?
(1) n = (a^4)(b^3) where a and b are different positive prime numbers.
(2) The only positive prime numbers that are factors of n are 5 and 7.
A.
We need to get the number of factors of positive integer n.
Statement 1=>
As a and b are "different" prime numbers => Number of factors of a must be 5*4=20
Hence sufficient .
Statement 2=>
There exist ∞ numbers with the same set of prime numbers.
E.g
5*7=> Four factors.
5^2*7^2=> Nine factors.
Etc.
Hence not sufficient.
Hence A.
We need to get the number of factors of positive integer n.
Statement 1=>
As a and b are "different" prime numbers => Number of factors of a must be 5*4=20
Hence sufficient .
Statement 2=>
There exist ∞ numbers with the same set of prime numbers.
E.g
5*7=> Four factors.
5^2*7^2=> Nine factors.
Etc.
Hence not sufficient.
Hence A.
174)Data Sufficiency->If x and y are positive integers and x + y = 3^x, is y divisible by 6?
(1) x is odd.
(2) x is a multiple of 3.
Given info->x and y are positive integers.
x+y=3^x
In order for y to be divisible by 6 => It must be divisible by both 2 and 3.
Statement 1->
Test Case 1->
x is odd
x=1
y=y
x+y=3^1=> Yes
Clearly y is not divisible by 6.
Test Case 2->
x =3
y=24
x+y=3^3=>Yes
Clearly y is divisible by 6
Hence not sufficient.
Statement 1->
x is a multiple of 3.
So y must be a multiple of 3 too.
Now=>
Test case 1->
x=3
y=24
x+y=3^3=>Yes
Clearly y is divisible by 6.
Test case 2->
x=6
y=3^6-6=>Odd number -Even number =>Odd number => Never divisible by 6.
Hence not sufficient.
Combing the two statements=>
From Statement 1=> y=> even
From Statement 2=> y must be a multiple of 3.
Hence y must be a multiple of 6.
Hence Sufficient.
Hence C.
x+y=3^x
In order for y to be divisible by 6 => It must be divisible by both 2 and 3.
Statement 1->
Test Case 1->
x is odd
x=1
y=y
x+y=3^1=> Yes
Clearly y is not divisible by 6.
Test Case 2->
x =3
y=24
x+y=3^3=>Yes
Clearly y is divisible by 6
Hence not sufficient.
Statement 1->
x is a multiple of 3.
So y must be a multiple of 3 too.
Now=>
Test case 1->
x=3
y=24
x+y=3^3=>Yes
Clearly y is divisible by 6.
Test case 2->
x=6
y=3^6-6=>Odd number -Even number =>Odd number => Never divisible by 6.
Hence not sufficient.
Combing the two statements=>
From Statement 1=> y=> even
From Statement 2=> y must be a multiple of 3.
Hence y must be a multiple of 6.
Hence Sufficient.
Hence C.
175)Data Sufficiency ->If x^3 – x = n and x is a positive integer greater than 1, is n divisible by 8?
(1) When 3x is divided by 2, there is a remainder.
(2) x = 4y + 1, where y is an integer.
D.
As n=x^3-x=> (x-1)*x*(x+1)=> If x is odd => both x-1 and x+1 will be consecutive even integers.So one of them would be multiple of 4 and other would be a multiple of 2 for sure.Hence n would be a multiple of 8.
Statement 1->
3x is odd => x is odd => Sufficient.
Statement 2->
x=4y+1 => Even+odd=odd
Hence Sufficient.
Hence D.
As n=x^3-x=> (x-1)*x*(x+1)=> If x is odd => both x-1 and x+1 will be consecutive even integers.So one of them would be multiple of 4 and other would be a multiple of 2 for sure.Hence n would be a multiple of 8.
Statement 1->
3x is odd => x is odd => Sufficient.
Statement 2->
x=4y+1 => Even+odd=odd
Hence Sufficient.
Hence D.
176)If a positive odd integer N has p positive factors, how many positive factors will 2N have ?
A) p
B) 2p
C) P+1
D) 2p+1
E) Cannot be determined
B.
177)Data Sufficiency->Is the integer n odd?
(1) n is divisible by 3
(2) 2n is divisible by twice as many positive integers as n
B.
We are told that n is an integer.
Statement 1->
n/3=integer.
E.g=> n=3 or n=6 etc.
Hence not sufficient.
Statement 2->
If n is odd => The number of factors of 2n will always be 2 times the number of factors of n.
This is because any odd number will just have odd divisors.It cannot have any even divisor as 2 is not its prime factor.
After we multiple 2 to n => The numb of factors will get doubled as for every odd factor there would be an even factor.
Hence n is odd
Thus-> This statement is sufficient.
Hence B.
We are told that n is an integer.
Statement 1->
n/3=integer.
E.g=> n=3 or n=6 etc.
Hence not sufficient.
Statement 2->
If n is odd => The number of factors of 2n will always be 2 times the number of factors of n.
This is because any odd number will just have odd divisors.It cannot have any even divisor as 2 is not its prime factor.
After we multiple 2 to n => The numb of factors will get doubled as for every odd factor there would be an even factor.
Hence n is odd
Thus-> This statement is sufficient.
Hence B.
178)If x is a prime number greater than 5, y is a positive integer, and 5y=x^2+x, then y must be divisible by which of the following?
I. 5
II. 2x
III. x+1
A. I only
B. II only
C. III only
D. I and II only
E. II and III only
B.
179)If P is a prime number greater than 5, what is the remainder when P^2 is divided by 8.
A) 4
B) 3
C) 2
D) 1
E) Cannot be determined
D.
All prime numbers greater than 2 are odd.
Hence p must be of the form => 2k+1
p^2=> 4k^2+4k+1=> 4k(k+1)+1= 8k'+1
Hence it will always leave a remainder 1 with 8.
Another important takeaway ->
For any odd number k => k^2 will always leave a remainder 1 with 8.
All prime numbers greater than 2 are odd.
Hence p must be of the form => 2k+1
p^2=> 4k^2+4k+1=> 4k(k+1)+1= 8k'+1
Hence it will always leave a remainder 1 with 8.
Another important takeaway ->
For any odd number k => k^2 will always leave a remainder 1 with 8.
180)Data Sufficiency->Is positive integer N divisible by 3?
(1) N^2/36 is an integer
(2) 144/N^2 is an integer
A.
A very useful property -> X and X^n always have the exact same prime factors.
For N to be divisible by 3 => 3 must be the prime factors of N.
Statement 1=>
N^2/36 is an integer.
N^2 has both 2 and 3 as it prime.
Thus N must have 3 as its prime too.
Hence Sufficient.
Statement 2=>
Lets use test Cases here.
N=1 => 144/1^2 =Integer => 1 is not divisible by 3.
N=3 => 144/N^2 =integer => 3 is divisible by 3.
Hence not sufficient.
Hence A.
A very useful property -> X and X^n always have the exact same prime factors.
For N to be divisible by 3 => 3 must be the prime factors of N.
Statement 1=>
N^2/36 is an integer.
N^2 has both 2 and 3 as it prime.
Thus N must have 3 as its prime too.
Hence Sufficient.
Statement 2=>
Lets use test Cases here.
N=1 => 144/1^2 =Integer => 1 is not divisible by 3.
N=3 => 144/N^2 =integer => 3 is divisible by 3.
Hence not sufficient.
Hence A.
181)Data Sufficiency->If x is an integer, is x^3 divisible by 7?
(1) 3*x^12 is divisible by 7
(2) 3*x^4 is divisible by 7
D.
Property in action =>X and X^n always have the exact same prime factors.
Property in action =>X and X^n always have the exact same prime factors.
182)Data Sufficiency->How many different prime factors does positive integer n have?
(1) 48< n^2 < 99
(2) 125n^2 has twelve factors
D.
183)Data Sufficiency->Is the integer k divisible by 6?
(1) 6k is divisible by 12.
(2) 9k is divisible by 12.
E.
We can take k=4 and k=4*6 to arrive at E.
We can take k=4 and k=4*6 to arrive at E.
184)If n = 3^8 - 2^8, which of the following is NOT a factor of n?
(A) 97
(B) 65
(C) 35
(D) 13
(E) 5
C.
Here => Using a^2-b^2=(a+b)*(a-b)=>3^8-2^8 = 97*13*5.
Clearly 35 is not a factor.
Hence C.
Source-> Official Guide.
Here => Using a^2-b^2=(a+b)*(a-b)=>3^8-2^8 = 97*13*5.
Clearly 35 is not a factor.
Hence C.
Source-> Official Guide.
185)Data Sufficiency->If n is a positive integer, is n^3 – n divisible by 24 ?
(1) n = 2k + 1, where k is an integer.
(2) n^2 + n is divisible by 6
A.
Firstly n^3-n= (n-1)n(n+1)
It is a product of 3 consecutive integers.
Regardless of what the value of n is => It will always be a multiple of 6.
Reason -> Product of n consecutive integers is divisible by n!
Now lets see two cases ->
Case 1-> n is odd
If n is odd => n-1 will be even and n+1 will be even too.
Further more one out of them will be multiple of 4.
Thus (n-1)n(n+1) will be multiple of 8.
As it is already a multiple of 6 => we can conclude that (n-1)n(n+1) will be a multiple of 24.
Case 2-> n is even
In that scenario it is not possible for us to conclude anything except (n-1)n(n+1) will be multiple of 6.
Hence we just need to see if n is odd or not.
Statement 1->
n=2k+1 for some integer k.
Hence n=even+odd=odd
Hence n must be a multiple of 24.
Hence sufficient.
Statement 2->
Lest use some test cases=>
n=2 => (n-1)n(n+1) = 1*2*3 => Not a multiple of 4.
n=3 => (n-1)n(n+1)=2*3*4 =>Multiple of 24.
Hence not sufficient.
Hence A.
Source->Official Guide.
Firstly n^3-n= (n-1)n(n+1)
It is a product of 3 consecutive integers.
Regardless of what the value of n is => It will always be a multiple of 6.
Reason -> Product of n consecutive integers is divisible by n!
Now lets see two cases ->
Case 1-> n is odd
If n is odd => n-1 will be even and n+1 will be even too.
Further more one out of them will be multiple of 4.
Thus (n-1)n(n+1) will be multiple of 8.
As it is already a multiple of 6 => we can conclude that (n-1)n(n+1) will be a multiple of 24.
Case 2-> n is even
In that scenario it is not possible for us to conclude anything except (n-1)n(n+1) will be multiple of 6.
Hence we just need to see if n is odd or not.
Statement 1->
n=2k+1 for some integer k.
Hence n=even+odd=odd
Hence n must be a multiple of 24.
Hence sufficient.
Statement 2->
Lest use some test cases=>
n=2 => (n-1)n(n+1) = 1*2*3 => Not a multiple of 4.
n=3 => (n-1)n(n+1)=2*3*4 =>Multiple of 24.
Hence not sufficient.
Hence A.
Source->Official Guide.
186)For which of the following values of n is (100+n)/n NOT an integer?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
C.
Source->Official Guide.
Source->Official Guide.
187)Data Sufficiency->Is a*b*c divisible by 24?
(1) a, b, and c are consecutive even integers
(2) a*b is divisible by 12
A.
We need to see if a*b*c is divisible by 24 or not.
Statement 1->
a,b,c are consecutive evens =>Let the consecutive evens be =>
2n
2n+2
2n+4
Taking the product => 2n(2n+2)(2n+4)=> 8n(n+1)(n+2)
Product of t consecutive integers is always divisible by t!
Hence 8n(n+1)(n+2)=8*6k = 48k for some integer k.
Clearly it will be divisible by 24.
Hence sufficient.
Statement 2->
Here we have no clue whether a,b,c are integers or non integers.
Hence not sufficient.
Hence A.
We need to see if a*b*c is divisible by 24 or not.
Statement 1->
a,b,c are consecutive evens =>Let the consecutive evens be =>
2n
2n+2
2n+4
Taking the product => 2n(2n+2)(2n+4)=> 8n(n+1)(n+2)
Product of t consecutive integers is always divisible by t!
Hence 8n(n+1)(n+2)=8*6k = 48k for some integer k.
Clearly it will be divisible by 24.
Hence sufficient.
Statement 2->
Here we have no clue whether a,b,c are integers or non integers.
Hence not sufficient.
Hence A.
188)Data Sufficiency->If x is the product of integers a, b, c, and d, is x divisible by 128?
(1) a = 24
(2) a, b, c, and d are consecutive even integers
B.
We are told that a,b,c,d are all integers.
We need to see if x=a*b*c*d is divisible by 128 or not.
Statement 1->
Lets use test cases.
Case 1=>
a=24
b=1
c=1
d=1
x=24 => Clearly it is not divisible by 128.
Case 2=>
a=24
b=1
c=1
d=128
x=24*128 => divisible by 128.
Hence not sufficient.
Statement 2->
Consecutive evens =>
a=2n
b=2n+2
c=2n+4
d=2n+6
Taking the Product => x=2n(2n+2)(2n+4)(2n+6)=> 8*n(n+1)(n+2)(n+3)
Property -> Product of n consecutive integers is always divisible by n!
Hence x=8*24k For some integer k=> 128k'
Hence x must be divisible by 128.
Hence sufficient.
Hence B.
We are told that a,b,c,d are all integers.
We need to see if x=a*b*c*d is divisible by 128 or not.
Statement 1->
Lets use test cases.
Case 1=>
a=24
b=1
c=1
d=1
x=24 => Clearly it is not divisible by 128.
Case 2=>
a=24
b=1
c=1
d=128
x=24*128 => divisible by 128.
Hence not sufficient.
Statement 2->
Consecutive evens =>
a=2n
b=2n+2
c=2n+4
d=2n+6
Taking the Product => x=2n(2n+2)(2n+4)(2n+6)=> 8*n(n+1)(n+2)(n+3)
Property -> Product of n consecutive integers is always divisible by n!
Hence x=8*24k For some integer k=> 128k'
Hence x must be divisible by 128.
Hence sufficient.
Hence B.
189)Data Sufficiency->How many positive factors does positive integer N have?
1) N^3 has Four positive factors
2) 5N has four positive factors
A.
From statement 1-> N must be a prime number.Hence it will always have 2 factors.Hence sufficient.
From statement 2 -> Two cases are possible => N=5^2 and N=Any prime number other than 5.
Hence N can have 3 or two factors.
Hence not sufficient.
Hence A.
From statement 1-> N must be a prime number.Hence it will always have 2 factors.Hence sufficient.
From statement 2 -> Two cases are possible => N=5^2 and N=Any prime number other than 5.
Hence N can have 3 or two factors.
Hence not sufficient.
Hence A.
190)Which of the following statements must be true?
A)If n is even then n^3-n is divisible by 24.
B)The sum of first 1000 prime numbers is even.
C)The product of first 32 primes is odd.
D)2 and 3 are the only consecutive prime numbers.
E)If n is odd then n^3-n is always divisible by 24.
E.
Mote-> D is not true as there is a BIG difference between consecutive prime numbers and consecutive numbers that are prime.
So 5,7 or 11,13 or 17,19 etc are all consecutive prime numbers.
2,3 are the only consecutive integers that are also prime numbers.
Mote-> D is not true as there is a BIG difference between consecutive prime numbers and consecutive numbers that are prime.
So 5,7 or 11,13 or 17,19 etc are all consecutive prime numbers.
2,3 are the only consecutive integers that are also prime numbers.
191)Which of the following is NOT a factor of 10! ?
A. 1440
B. 625
C. 160
D. 80
E. 50
B.
Number of 5's in 10!=2
So 625 which has five 5's can never be a factor of 10!
Hence B.
Number of 5's in 10!=2
So 625 which has five 5's can never be a factor of 10!
Hence B.
192)How many prime factors does the positive integer N have?
A)N^3 has 5 prime factors.
B)10N^3 has 7 Prime factors.
A.
193)Data Sufficiency->How many prime factors does N have?
(1) N is a factor of 7200.
(2) 180 is a factor of N.
C.
We need to get the number of prime factors of N.
Statement 1->
N is a factor of 7200
7200=2^5*3^2*5
Hence N can have -->
Zero prime factor if N=1
One prime factor
Two prime factors
Or Three Prime factors.
Hence not sufficient.
Statement 2->
180 is a factor of N.
Hence N=2^2*3^2*5*k for some integer K.
Hene N must have 2,3,5 as its prime factors.
Other than that N can have any Prime factors ≥3
Hence not sufficient.
Combining the two statements =>
N must have exactly three prime factors>
Hence sufficient.
Hence C.
We need to get the number of prime factors of N.
Statement 1->
N is a factor of 7200
7200=2^5*3^2*5
Hence N can have -->
Zero prime factor if N=1
One prime factor
Two prime factors
Or Three Prime factors.
Hence not sufficient.
Statement 2->
180 is a factor of N.
Hence N=2^2*3^2*5*k for some integer K.
Hene N must have 2,3,5 as its prime factors.
Other than that N can have any Prime factors ≥3
Hence not sufficient.
Combining the two statements =>
N must have exactly three prime factors>
Hence sufficient.
Hence C.
194)Data Sufficiency->If F is the prime factorization of N!, how many factors in F have an exponent of 1?
(1) 30 ≤ N ≤ 40
(2) 25 ≤ N ≤ 35
E.
195)Data Sufficiency->Given that \(x\) is an integer and \(x\) is positive, is \(\frac{210}{x}\) also an integer??
(1) \(x\) is a prime number.
(2) \(x\) < 8
C.
Observe that 210=2*3*5*7
We need to see if 210/x is an integer or not.
Statement 1->
x is prime
E.g=> x=13 => No
x=7 => Yes
Hence not sufficient.
Statement 2->
x<8
E.g-> x=4 -> No
x=7 => Yes
Hence not sufficient.
combining the two statements =>
x can be 2,3,5,7
Each one if them is a factor of 210.
Hence 210/x will always be an integer.
Hence sufficient.
Hence C.
Observe that 210=2*3*5*7
We need to see if 210/x is an integer or not.
Statement 1->
x is prime
E.g=> x=13 => No
x=7 => Yes
Hence not sufficient.
Statement 2->
x<8
E.g-> x=4 -> No
x=7 => Yes
Hence not sufficient.
combining the two statements =>
x can be 2,3,5,7
Each one if them is a factor of 210.
Hence 210/x will always be an integer.
Hence sufficient.
Hence C.
196)Which of the following is NOT a factor of the product of the first 50 positive multiples of 4 ?
A. \(17^2\)
B. \(11^4\)
C. \(7^6\)
D. \(47^{12}\)
E. \(2^{124}\)
D.
Firstly the product of first positive 50 multiples of 4=>
4*1*4*2*4*3*4*4*4*5.....4*50=> 4^50*50!
Secondly notice Option D is 47^2 and 47 is a prime number.
Number of 47's in 4^50*50!=> one
Hence 47^2 will never be its factor.
Hence D.
Firstly the product of first positive 50 multiples of 4=>
4*1*4*2*4*3*4*4*4*5.....4*50=> 4^50*50!
Secondly notice Option D is 47^2 and 47 is a prime number.
Number of 47's in 4^50*50!=> one
Hence 47^2 will never be its factor.
Hence D.
197)Data Sufficiency->Is the integer \(n\) odd?
(1) a is an integer and \(n=a^7+a^5+a^3+a^2+2a+1\)
(2) 2n is divisible by twice as many positive integers as n
D.
Statement 1->
Here a can be even or odd => n will always be odd.
Hence sufficient.
Statement 2->
If n is odd => The number of factors of 2n will always be twice the number of factors of n.
Hence n is odd => Sufficient.
Hence D.
Statement 1->
Here a can be even or odd => n will always be odd.
Hence sufficient.
Statement 2->
If n is odd => The number of factors of 2n will always be twice the number of factors of n.
Hence n is odd => Sufficient.
Hence D.
198)If a positive integer n has 211 factors,then how many prime factors does n have?
A)one
B)two
C)three
D)Four
E)cannot be determined
A.
Notice here that 211 is a prime number as it is not divisile by any prime factor less than or equal to the square root of 211.
Therefore the number n can only be of the form => Prime^210
Hence it will always have one prime factor.
If the number of factors of any integer is a prime number => The integer will always have a single prime factor.
Hence A
Notice here that 211 is a prime number as it is not divisile by any prime factor less than or equal to the square root of 211.
Therefore the number n can only be of the form => Prime^210
Hence it will always have one prime factor.
If the number of factors of any integer is a prime number => The integer will always have a single prime factor.
Hence A
199)If a positive integer N has p factors ; how many factors will 2N have ?
A) p
B) 2p
C) P+1
D) 2p+1
E) Cannot be determined
E.
200)Data Sufficiency ->Is integer k a prime number?
(1) k = 10! + m, where 1 < m < 8
(2) k is a multiple of 7
A.
Note-->PM me if you find any answers/solutions inaccurate/inadequate.
Any Feedback would be Appreciated.
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